1 Introduction

The Split-\(\widehat{R}\) statistic and the effective sample size (abbreviated as ESS or \(S_{\rm eff}\); previously called \(N_{\rm eff}\) or \(n_{\rm eff}\)) are routinely used to monitor the convergence of iterative simulations, which are omnipresent in Bayesian statistics in the form of Markov-Chain Monte-Carlo samples. The original \(\widehat{R}\) statistic (Gelman and Rubin, 1992; Brooks and Gelman, 1998) and split-\(\widehat{R}\) (Gelman et al., 2013) are both based on the ratio of between and within-chain marginal variances of the simulations, while the latter is computed from split chains (hence the name).

\(\widehat{R}\), split-\(\widehat{R}\), and \(S_{\rm eff}\) are well defined only if the marginal distributions have finite mean and variance. Even if that’s the case, their estimates are less stable for distributions with long tails. To alleviate these problems, we define split-\(\widehat{R}\) and \(S_{\rm eff}\) using rank normalized values, empirical cumulative density functions, and small posterior intervals.

The code for the new split-\(\widehat{R}\) and \(S_{\rm eff}\) versions and for the corresponding diagnostic plots can be found in monitornew.R and monitorplot.R, respectively.

2 Review of split-\(\widehat{R}\) and effective sample size

In this section, we will review the split-\(\widehat{R}\) and effective sample size estimates as implemented in Stan 2.18 (Stan Development Team, 2018e). These implementations represent the current de facto standard of convergence diagnostics for iterative simulations.

2.1 Split-\(\widehat{R}\)

Below, we present the computation of Split-\(\widehat{R}\) following Gelman et al. (2013), but using the notation style of Stan Development Team (2018a). Our general recommendation is to always run several chains. \(N\) is the number of draws per chain, \(M\) is the number of chains, and \(S=MN\) is the total number of draws from all chains. For each scalar summary of interest \(\theta\), we compute \(B\) and \(W\), the between- and within-chain variances:

\[ B = \frac{N}{M-1}\sum_{m=1}^{M}(\overline{\theta}^{(.m)}-\overline{\theta}^{(..)})^2, \;\mbox{ where }\;\;\overline{\theta}^{(.m)}=\frac{1}{N}\sum_{n=1}^N \theta^{(nm)},\;\; \;\;\overline{\theta}^{(..)} = \frac{1}{M}\sum_{m=1}^M\overline{\theta}^{(.m)} \\ W = \frac{1}{M}\sum_{m=1}^{M}s_j^2, \;\mbox{ where }\;\; s_m^2=\frac{1}{N-1} \sum_{n=1}^N (\theta^{(nm)}-\overline{\theta}^{(.m)})^2. \]

The between-chain variance, \(B\), also contains the factor \(N\) because it is based on the variance of the within-chain means, \(\overline{\theta}^{(.m)}\), each of which is an average of \(N\) values \(\theta^{(nm)}\).

We can estimate \(\mbox{var}(\theta \mid y)\), the marginal posterior variance of the estimand, by a weighted average of \(W\) and \(B\), namely \[ \widehat{\mbox{var}}^+(\theta \mid y)=\frac{N-1}{N}W + \frac{1}{N}B. \] This quantity overestimates the marginal posterior variance assuming the starting distribution of the simulations is appropriately overdispersed compared to the target distribution, but is unbiased under stationarity (that is, if the starting distribution equals the target distribution), or in the limit \(N\rightarrow\infty\). To have an overdispersed starting distribution, independent Markov chains should be initialized with diffuse starting values for the parameters.

Meanwhile, for any finite \(N\), the within-chain variance \(W\) should underestimate \(\mbox{var}(\theta \mid y)\) because the individual chains haven’t had the time to explore all of the target distribution and, as a result, will have less variability. In the limit as \(N\rightarrow\infty\), the expectation of \(W\) also approaches \(\mbox{var}(\theta \mid y)\).

We monitor convergence of the iterative simulations to the target distribution by estimating the factor by which the scale of the current distribution for \(\theta\) might be reduced if the simulations were continued in the limit \(N\rightarrow\infty\). This potential scale reduction is estimated as \[ \widehat{R}= \sqrt{\frac{\widehat{\mbox{var}}^+(\theta \mid y)}{W}}, \] which declines to 1 as \(N\rightarrow\infty\). We call this split-\(\widehat{R}\) because we are applying it to chains that have been split in half so that \(M\) is twice the number of actual chains. Without splitting, \(\widehat{R}\) would get fooled by non-stationary chains (see Appendix D).

We note that split-\(\widehat{R}\) is also well defined for sequences that are not Markov-chains. However, for simplicity, we always refer to ‘chains’ instead of more generally to ‘sequences’ as the former is our primary use case for \(\widehat{R}\)-like measures.

2.2 Effective sample size

If the \(N\) simulation draws within each chain were truly independent, the between-chain variance \(B\) would be an unbiased estimate of the posterior variance, \(\mbox{var}(\theta \mid y)\), and we would have a total of \(S = MN\) independent simulations from the \(M\) chains. In general, however, the simulations of \(\theta\) within each chain will be autocorrelated, and thus \(B\) will be larger than \(\mbox{var}(\theta \mid y)\), in expectation.

A nice introductory reference for analyzing MCMC results in general and effective sample size in particular is provided by Geyer (2011, see also 1992). The particular calculations used by Stan (Stan Development Team, 2018e) follow those for split-\(\widehat{R}\), which involve both between-chain (mean) and within-chain calculations (autocorrelation). They were introduced in the Stan manual (Stan Development Team, 2018d) and explained in more detail in Gelman et al. (2013).

One way to define effective sample size for correlated simulation draws is to consider the statistical efficiency of the average of the simulations \(\bar{\theta}^{(..)}\) as an estimate of the posterior mean \(\mbox{E}(\theta \mid y)\). This generalizes also to posterior expectations of functionals of parameters \(\mbox{E}(g(\theta) \mid y)\) and we return later to how to estimate the effective sample size of quantiles which cannot be presented as expectations. For simplification, in this section we consider the effective sample size for the posterior mean.

The effective sample size of a chain is defined in terms of the autocorrelations within the chain at different lags. The autocorrelation \(\rho_t\) at lag \(t \geq 0\) for a chain with joint probability function \(p(\theta)\) with mean \(\mu\) and variance \(\sigma^2\) is defined to be \[ \rho_t = \frac{1}{\sigma^2} \, \int_{\Theta} (\theta^{(n)} - \mu) (\theta^{(n+t)} - \mu) \, p(\theta) \, d\theta. \] This is just the correlation between the two chains offset by \(t\) positions. Because we know \(\theta^{(n)}\) and \(\theta^{(n+t)}\) have the same marginal distribution in an MCMC setting, multiplying the two difference terms and reducing yields \[ \rho_t = \frac{1}{\sigma^2} \, \int_{\Theta} \theta^{(n)} \, \theta^{(n+t)} \, p(\theta) \, d\theta. \]

The effective sample size of one chain generated by a process with autocorrelations \(\rho_t\) is defined by \[ N_{\rm eff} \ = \ \frac{N}{\sum_{t = -\infty}^{\infty} \rho_t} \ = \ \frac{N}{1 + 2 \sum_{t = 1}^{\infty} \rho_t}. \]

Effective sample size \(N_{\rm eff}\) can be larger than \(N\) in case of antithetic Markov chains, which have negative autocorrelations on odd lags. The Dynamic Hamiltonian Monte-Carlo algorithms used in Stan (Hoffman and Gelman, 2014; Betancourt, 2017) can produce \(N_{\rm eff}>N\) for parameters with a close to Gaussian posterior (in the unconstrained space) and low dependency on other parameters.

2.2.1 Estimation of the Effective Sample Size

In practice, the probability function in question cannot be tractably integrated and thus neither autocorrelation nor the effective sample size can be calculated. Instead, these quantities must be estimated from the samples themselves. The rest of this section describes an autocorrelation and split-\(\widehat{R}\) based effective sample size estimator, based on multiple split chains. For simplicity, each chain will be assumed to be of the same length \(N\).

Stan carries out the autocorrelation computations for all lags simultaneously using Eigen’s fast Fourier transform (FFT) package with appropriate padding; see Geyer (2011) for more details on using FFT for autocorrelation calculations. The autocorrelation estimates \(\hat{\rho}_{t,m}\) at lag \(t\) from multiple chains \(m \in (1,\ldots,M)\) are combined with the within-chain variance estimate \(W\) and the multi-chain variance estimate \(\widehat{\mbox{var}}^{+}\) introduced in the previous section to compute the combined autocorrelation at lag \(t\) as \[ \hat{\rho}_t = 1 - \frac{\displaystyle W - \textstyle \frac{1}{M}\sum_{m=1}^M \hat{\rho}_{t,j}}{\widehat{\mbox{var}}^{+}}. \label{rhohat} \] If the chains have not converged, the variance estimator \(\widehat{\mbox{var}}^{+}\) will overestimate the true marginal variance which leads to an overestimation of the autocorrelation and an underestimation of the effective sample size.

Because of noise in the correlation estimates \(\hat{\rho}_t\) increases as \(t\) increases, typically the truncated sum of \(\hat{\rho}_t\) is used. Negative autocorrelations can happen only on odd lags and by summing over pairs starting from lag \(t=0\), the paired autocorrelation is guaranteed to be positive, monotone and convex modulo estimator noise (Geyer, 1992, 2011). Stan 2.18 uses Geyer’s initial monotone sequence criterion. The effective sample size of combined chains is defined as \[ S_{\rm eff} = \frac{N \, M}{\hat{\tau}}, \] where \[ \hat{\tau} = 1 + 2 \sum_{t=1}^{2k+1} \hat{\rho}_t = -1 + 2 \sum_{t'=0}^{k} \hat{P}_{t'}, \] and \(\hat{P}_{t'}=\hat{\rho}_{2t'}+\hat{\rho}_{2t'+1}\). The initial positive sequence estimator is obtained by choosing the largest \(k\) such that \(\hat{P}_{t'}>0\) for all \(t' = 1,\ldots,k\). The initial monotone sequence estimator is obtained by further reducing \(\hat{P}_{t'}\) to the minimum of the preceding values so that the estimated sequence becomes monotone.

The effective sample size \(S_{\rm eff}\) described here is different from similar formulas in the literature in that we use multiple chains and between-chain variance in the computation, which typically gives us more conservative claims (lower values of \(S_{\rm eff}\)) compared to single chain estimates, especially when mixing of the chains is poor. If the chains are not mixing at all (e.g., the posterior is multimodal and the chains are stuck in different modes), then our \(S_{\rm eff}\) is close to the number of chains.

Before version 2.18, Stan used a slightly incorrect initial sequence which implied that \(S_{\rm eff}\) was capped at \(S\) and thus the effective sample size was underestimated for some models. As antithetic behavior (i.e., \(S_{\rm eff} > S\)) is not that common or the effect is small, and capping at \(S\) can be considered to be pessimistic, this had negligible effect on any inference. However, it may have led to an underestimation of Stan’s efficiency when compared to other packages performing MCMC sampling.

3 Rank normalized split-\(\widehat{R}\) and effective sample size

As split-\(\widehat{R}\), and \(S_{\rm eff}\) are well defined only if the marginal posteriors have finite mean and variance, we next introduce split-\(\widehat{R}\) and \(S_{\rm eff}\) using rank normalized values, empirical cumulative density functions, and small posterior intervals which are well defined for all distributions and more robust for long tailed distributions.

3.1 Rank normalized split-\(\widehat{R}\)

Rank normalized split-\(\widehat{R}\) is computed using the equations in Section Split-\(\widehat{R}\) by replacing the original parameter values \(\theta^{(nm)}\) with their corresponding rank normalized values \(z^{(nm)}\).

Rank normalization:

Rank: Replace each value \(\theta^{(nm)}\) by its rank \(r^{(nm)}\). Average rank for ties are used to conserve the number of unique values of discrete quantities. Ranks are computed jointly for all draws from all chains.
Normalize: Normalize ranks by inverse normal transformation \(z^{(nm)} = \phi^{-1}((r^{(nm)}-1/2)/S)\).

Appendix B illustrates the rank normalization of multiple chains.

For continuous variables and \(S \rightarrow \infty\), the rank normalized values are normally distributed and rank normalization performs non-parametric transformation to normal distribution. Using normalized ranks instead of ranks directly, has the benefit that the behavior of \(\widehat{R}\) and \(S_{\rm eff}\) do not change from the ones presented in Section Split-\(\widehat{R}\) for normally distributed \(\theta\).

3.2 Rank normalized folded-split-\(\widehat{R}\)

Both original and rank-normalized split-\(\widehat{R}\) can be fooled if chains have different scales but the same location (see Appendix D). To alleviate this problem, we propose to compute rank normalized folded-split-\(\widehat{R}\) using folded split chains by rank normalizing absolute deviations from median \[ {\rm abs}(\theta^{(nm)}-{\rm median}(\theta)). \]

To obtain a single conservative \(\widehat{R}\) estimate, we propose to report the maximum of rank normalized split-\(\widehat{R}\) and rank normalized folded-split-\(\widehat{R}\) for each parameter.

3.3 Effective sample size using rank normalized values

In addition to using rank-normalized values for convergence diagnostics via \(\widehat{R}\), we can also compute the corresponding effective sample size \(S_{\rm eff}\), using equations in Section Effective sample size by replacing parameter values \(\theta^{(nm)}\) with rank normalized values \(z^{(nm)}\). This estimate will be well defined even if the original distribution does not have finite mean and variance. It is not directly applicable to estimate the Monte Carlo error of the mean of the original values, but it will provide a bijective transformation-invariant estimate of the mixing efficiency of chains. For parameters with a close to normal distribution, the difference to using the original values is small. However, for parameters with a distribution far from normal, rank normalization can be seen as a near optimal non-parametric transformation.

The effective sample size using rank normalized values is a useful measure for efficiency of estimating the bulk (mean and quantiles near the median) of the distribution, and as shorthand term we use term bulk effective sample size (bulk-ESS). Bulk-ESS is also useful for diagnosing problems due to trends or different means of the chains (see Appendix D).

3.4 Efficiency estimates for the cumulative distribution function

The bulk- and tail-ESS introduced above are useful as overall efficiency measures. Next, we introduce effective sample size estimates for the cumulative distribution function (CDF), and later we use that to introduce efficiency diagnostics for quantiles and local small probability intervals.

Quantiles and posterior intervals derived on their basis are often reported quantities which are easy to estimate from posterior draws. Estimating the efficiency of such quantile estimates thus has a high practical relevance in particular as we observe the efficiency for tail quantiles to often be lower than for the mean or median. The \(\alpha\)-quantile is defined as the parameter value \(\theta_\alpha\) for which \(p(\theta \leq \theta_\alpha) = \alpha\). An estimate \(\hat{\theta}_\alpha\) of \(\theta_\alpha\) can thus be obtained by finding the \(\alpha\)-quantile of the empirical CDF (ECDF) of the posterior draws \(\theta^{(s)}\). However, quantiles cannot be written as an expectation, and thus the above equations for \(\widehat{R}\) and \(S_{\rm eff}\) are not directly applicable. Thus, we first focus on the efficiency estimate for the cumulative probability \(p(\theta \leq \theta_\alpha)\) for different values of \(\theta_\alpha\).

For any \(\theta_\alpha\), the ECDF gives an estimate of the cumulative probability \[ p(\theta \leq \theta_\alpha) \approx \bar{I}_\alpha = \frac{1}{S}\sum_{s=1}^S I(\theta^{(s)} \leq\theta_\alpha), \] where \(I()\) is the indicator function. The indicator function transforms simulation draws to 0’s and 1’s, and thus the subsequent computations are bijectively invariant. Efficiency estimates of the ECDF at any \(\theta_\alpha\) can now be obtained by applying rank-normalizing and subsequent computations directly on the indicator function’s results. See Appendix C for an illustration of variance of ECDF.

3.5 Efficiency estimates for quantiles and tail-ESS

Assuming that we know the CDF to be a certain continuous function \(F\) which is smooth near an \(\alpha\)-quantile of interest, we could use the delta method to compute a variance estimate for \(F^{-1}(\bar{I}_\alpha)\). Although we don’t usually know \(F\), the delta method approach reveals that the variance of \(\bar{I}_\alpha\) for some \(\theta_\alpha\) is scaled by the (usually unknown) density \(f(\theta_\alpha)\), but the efficiency depends only on the efficiency of \(\bar{I}_\alpha\). Thus, we can use the effective sample size for the ECDF (we computed using the indicator function \(I(\theta^{(s)} \leq \theta_\alpha)\)) also for the corresponding quantile estimates.

In order to summarize the efficiency in the distributions’ tails, we propose to compute the minimum of the effectve sample sizes of the 5% and 95% quantiles. As a shorthand, we use the term tail effective sample size (tail-ESS). Tail-ESS is also useful for diagnosing problems due to different scales of the chains (see Appendix D).

3.6 Efficiency estimates for median and MAD

Since the marginal posterior distributions might not have finite mean and variance, by default RStan (Stan Development Team, 2018c) and RStanARM (Stan Development Team, 2018b) report median and median absolute deviation (MAD) instead of mean and standard error (SE). Median and MAD are well defined even when the marginal distribution does not have finite mean and variance. Since the median is just 50%-quantile, we can get efficiency estimate for it as for any other quantile.

We can also compute the efficiency estimate for the median absolute deviation (MAD), by computing the efficiency estimate for the median of absolute deviations from the median of all draws. The absolute deviations from the median of all draws are same as previously defined for folded samples \[ {\rm abs}(\theta^{(nm)}-{\rm median}(\theta)). \] We see that the efficiency estimate for MAD is obtained by using the same approach as for the median (and other quantiles) but with the folded values also used in rank-normalized-folded-split-\(S_{\rm eff}\).

3.7 Monte Carlo error estimates for quantiles

Previously, Stan has reported Monte Carlo standard error estimates for the mean of a quantity. This is valid only if the corresponding marginal distribution has finite mean and variance; and even if valid, it may be easier and more robust to focus on the median and other quantiles, instead.

Median, MAD and quantiles are well defined even when the distribution does not have finite mean and variance, and they are asymptotically normal for continuous distributions which are non-zero in the relevant neighborhood. As the delta method for computing the variance would require explicit knowledge of the normalized posterior density, we propose the following alternative approach:

Compute quantiles of the \({\rm Beta}(S_{\rm eff}/S \bar{I}_\alpha+1, S_{\rm eff}/S(1-\bar{I}_\alpha)+1)\) distribution. Including \(S_{\rm eff}/S\) takes into account the efficiency estimate for the posterior draws.
Find indices in \(\{1,\ldots,S\}\) closest to the ranks of these quantiles. For example, for quantile \(Q\), find \(s = {\rm round(Q S)}\).
Use the corresponding \(\theta^{(s)}\) from the list of sorted posterior draws as quantiles from the error distribution. These quantiles can be used to approximate the Monte Carlo standard error.

3.8 Efficiency estimate for small interval probability estimates

We can get more local efficiency estimates by considering small probability intervals. We propose to compute the efficiency estimates for \[ \bar{I}_{\alpha,\delta} = p(\hat{Q}_\alpha < \theta \leq \hat{Q}_{\alpha+\delta}), \] where \(\hat{Q}_\alpha\) is an empirical \(\alpha\)-quantile, \(\delta=1/k\) is the length of the interval with some positive integer \(k\), and \(\alpha \in (0,\delta,\ldots,1-\delta)\) changes in steps of \(\delta\). Each interval has \(S/k\) draws, and the efficiency measures autocorrelation of an indicator function which is \(1\) when the values are inside the specific interval and \(0\) otherwise. This gives us a local efficiency measure which does not depend on the shape of the distribution.

3.9 Rank plots

In addition to using rank-normalized values to compute split-\(\widehat{R}\), we propose to use rank plots for each chain instead of trace plots. Rank plots are nothing else than histograms of the ranked posterior samples (ranked over all chains) plotted separately for each chain. If rank plots of all chains look similar, this indicates good mixing of the chains. As compared to trace plots, rank plots don’t tend to squeeze to a mess in case of long chains.

3.10 Proposed changes in Stan

The proposal is to switch in Stan:

from split-\(\widehat{R}\) to the maximum of rank-normalized-split-\(\widehat{R}\) and rank-normalized-folded-split-\(\widehat{R}\)
from the classic effective sample size estimate (Neff), which currently doesn’t use chain splitting, to rank-normalized-split-\(S_{\rm eff}\) (bulk-ESS) and rank-normalized-folded-split-\(S_{\rm eff}\) (tail-ESS)

Justifications for the changes are:

Rank normalization makes \(\widehat{R}\) and effective sample size measures well defined for all distributions, invariant under bijective transformations, and more stable than their classical counterparts.
Adding folded versions of \(\widehat{R}\) and effective sample size estimates helps in detecting scale differences across chains.

In summary outputs, we propose to use Rhat to denote also the new version. As \(n\) and \(N\) often refer to the number of observations, we propose to use acronym ESS for the effective sample size.

3.11 Proposed additions to bayesplot

We propose to add to the bayesplot package:

Rank plots
Plots of efficiency estimates for quantiles
Plots of efficiency estimates for small probability intervals
Plots of changes in efficiency estimates with increasing number of draws

3.12 Warning thresholds

Based on the experiments presented in Appendices D-F, more strict convergence diagnostics and effective sample size warning limits could be used. We propose the following warning thresholds although additional experiments would be useful:

Rhat > 1.01
ESS < 400

In case of running 4 chains, an effective sample size of 400 corresponds to having an effective sample size of 50 for each 8 split chains, which we consider to be minimum for reliable mean, variance and autocorrelation estimates needed for the convergence diagnostic. We recommend running at least 4 chains to get reliable between chain variances for the convergence diagnostics.

Plots shown in the upcoming sections have dashed lines based on these thresholds (in continuous plots, a dashed line at 1.005 is plotted instead of 1.01, as values larger than that are usually rounded in our summaries to 1.01).

4 Examples

In this section, we will go through some examples to demonstrate the usefulness of our proposed methods as well as the associated workflow in determining convergence. Appendices D-G contain more detailed analysis of different algorithm variants and further examples.

First, we load all the necessary R packages and additional functions.

library(tidyverse)
library(gridExtra)
library(rstan)
options(mc.cores = parallel::detectCores())
rstan_options(auto_write = TRUE)
library(bayesplot)
theme_set(bayesplot::theme_default(base_family = "sans"))
library(rjags)
library(abind)
source('monitornew.R')
source('monitorplot.R')

4.1 Cauchy: A distribution with infinite mean and variance

The classic split-\(\widehat{R}\) is based on calculating within and between chain variances. If the marginal distribution of a chain is such that the variance is not defined (i.e., infinite), the classic split-\(\widehat{R}\) is not well justified. In this section, we will use the Cauchy distribution as an example of such a distribution. Appendix E contains more detailed analysis of different algorithm variants and further Cauchy examples.

The following Cauchy models are from Michael Betancourt’s case study Fitting The Cauchy Distribution

4.1.1 Nominal parameterization of Cauchy

The nominal Cauchy model with direct parameterization is as follows:

writeLines(readLines("cauchy_nom.stan"))

parameters {
  vector[50] x;
}

model {
  x ~ cauchy(0, 1);
}

generated quantities {
  real I = fabs(x[1]) < 1 ? 1 : 0;
}

4.1.1.1 Default Stan options

Run the nominal model:

fit_nom <- stan(file = 'cauchy_nom.stan', seed = 7878, refresh = 0)

Warning: There were 1233 transitions after warmup that exceeded the maximum treedepth. Increase max_treedepth above 10. See
http://mc-stan.org/misc/warnings.html#maximum-treedepth-exceeded

Warning: Examine the pairs() plot to diagnose sampling problems

Treedepth exceedence and Bayesian fraction of missing information are dynamic HMC specific diagnostics (Betancourt, 2017). We get warnings about a very large number of transitions after warmup that exceeded the maximum treedepth, which is likely due to very long tails of the Cauchy distribution. All chains have low estimated Bayesian fraction of missing information also indicating slow mixing.

mon <- monitor(fit_nom)
print(mon)

Inference for the input samples (4 chains: each with iter = 2000; warmup = 1000):

         Q5      Q50   Q95    Mean    SD  Rhat Bulk_ESS Tail_ESS
x[1]   -5.7 -1.5e-02   6.3   2.500  36.0  1.03     1181      393
x[2]   -5.8 -1.2e-02   6.1   0.650  16.0  1.01     2645      502
x[3]   -5.2  8.7e-03   5.7   0.580  18.0  1.01     2683      823
x[4]   -6.2 -2.0e-02   6.9   0.160  11.0  1.01     3627      644
x[5]   -9.7 -4.8e-02   5.1  -1.400  11.0  1.01      629      156
x[6]   -5.3 -3.4e-03   5.4   0.200   5.3  1.01     3060      883
x[7]   -6.4  5.9e-02  11.0   4.100  33.0  1.02      607      184
x[8]   -6.4 -1.2e-02   5.4  -0.220   7.7  1.00     2658      886
x[9]   -6.5 -3.5e-03   6.3   0.150   7.4  1.01     3128      901
x[10]  -6.1 -1.1e-02   5.9   0.056   5.7  1.01     2421      642
x[11]  -6.7  1.5e-03   6.1   0.034   9.7  1.01     2079      600
x[12]  -5.7 -3.3e-02   4.9   0.190  18.0  1.00     2633      774
x[13]  -4.5 -5.5e-02   4.3   0.090   6.3  1.00     3148      811
x[14]  -4.9 -2.3e-03   5.0  -0.023   5.9  1.00     1461      492
x[15] -14.0 -5.5e-03  12.0  -1.400  23.0  1.03      486      160
x[16]  -7.0  1.3e-02   7.0   0.160  16.0  1.01     2329      463
x[17]  -6.6  1.2e-02   7.7   0.960  30.0  1.01     2292      446
x[18]  -4.5  5.4e-02   6.9   1.100  12.0  1.01     2640      447
x[19]  -7.7 -4.7e-02   6.1  -3.100  28.0  1.01     1147      298
x[20]  -5.8  2.6e-02  11.0   5.800  37.0  1.03      363       80
x[21]  -4.9  2.2e-02   5.4   0.110   5.5  1.01     3276      824
x[22]  -5.6  4.2e-02   5.4   0.490  15.0  1.01     2121      522
x[23] -14.0  7.6e-03   7.0  -3.100  32.0  1.01      391       89
x[24]  -5.9 -3.6e-02   5.5  -1.000  17.0  1.02     1434      284
x[25]  -7.1 -2.3e-02   5.9  -1.800  21.0  1.01     1544      324
x[26]  -9.0 -6.2e-02   5.8  -2.000  17.0  1.01     1778      452
x[27]  -8.8 -5.8e-03   8.3   0.150  18.0  1.00     1816      352
x[28]  -5.3  2.4e-02   5.9   0.110   9.5  1.02     3776      675
x[29]  -5.9 -2.4e-03   5.9  -0.036  18.0  1.01     3642      846
x[30]  -5.6 -1.6e-02   5.1  -0.180   6.2  1.00     4363      643
x[31]  -7.4  1.4e-02   7.3   0.069   8.7  1.00     3384      896
x[32]  -8.9 -3.8e-02   6.5 -10.000  81.0  1.01      561      141
x[33]  -4.8  3.1e-02   5.0  -0.350   8.6  1.01     2626      735
x[34]  -5.9 -3.4e-02   5.4  -0.110   6.0  1.01     2408      634
x[35]  -6.1  1.7e-02   6.8  -0.330  15.0  1.01     2654      630
x[36]  -4.9  9.7e-02  16.0  19.000 110.0  1.04      155       34
x[37]  -8.9 -9.1e-04   7.9  -0.090  13.0  1.01     1155      427
x[38]  -5.5 -2.3e-02   4.8  -0.330   6.0  1.00     3353      576
x[39]  -5.9 -2.7e-02   5.8   0.110   7.3  1.02     4493      724
x[40]  -8.7  1.2e-02   6.7  -2.000  20.0  1.00      877      148
x[41]  -6.9 -2.7e-02   7.5  -0.270  11.0  1.00     1726      512
x[42]  -6.2  1.4e-02   6.5   0.091  16.0  1.01     1519      448
x[43]  -6.4 -4.2e-04   7.4   0.110  12.0  1.01     2746      483
x[44]  -7.8  4.5e-03   8.0   0.410  11.0  1.01     2999      659
x[45]  -4.8 -2.0e-02   4.7   0.036   6.7  1.01     2904     1014
x[46]  -4.8  4.0e-03   6.0   1.200  22.0  1.00      955      338
x[47]  -8.5  3.4e-02  24.0   0.800  37.0  1.07      231       56
x[48]  -6.7  1.3e-03   5.3  -0.720  10.0  1.01     1907      469
x[49]  -6.3  4.4e-02   6.9   0.730  12.0  1.00     1490      390
x[50]  -5.2  8.3e-04   4.6  -0.220   7.1  1.00     3109      680
I       0.0  5.0e-01   1.0   0.500   0.5  1.00      390     4000
lp__  -93.0 -6.9e+01 -49.0 -70.000  13.0  1.05      117      323

For each parameter, Bulk_ESS and Tail_ESS are crude measures of 
effective sample size for bulk and tail quantities respectively (good values is 
ESS > 400), and Rhat is the potential scale reduction factor on rank normalized
split chains (at convergence, Rhat = 1).

which_min_ess <- which.min(mon[1:50, 'Tail_ESS'])

Several Rhat > 1.01 and some ESS < 400 indicate also that the results should not be trusted. The Appendix E has more results with longer chains as well.

We can further analyze potential problems using local efficiency and rank plots. We specifically investigate x[36], which has the smallest taill-ESS of 34.

We examine the sampling efficiency in different parts of the posterior by computing the efficiency of small interval probability estimates (see Section Efficiency estimate for small interval probability estimates). Each interval contains \(1/k\) of the draws (e.g., \(5\%\) each, if \(k=20\)). The small interval efficiency measures mixing of an function which indicates when the values are inside or outside the specific small interval. As detailed above, this gives us a local efficiency measure which does not depend on the shape of the distribution.

plot_local_ess(fit = fit_nom, par = which_min_ess, nalpha = 20)

We see that the efficiency of our posterior draws is worryingly low in the tails (which is caused by slow mixing in long tails of Cauchy). Orange ticks show iterations that exceeded the maximum treedepth.

An alternative way to examine the efficiency in different parts of the posterior is to compute efficiencies estimates for quantiles (see Section Efficiency for quantiles). Each interval has a specified proportion of draws, and the efficiency measures mixing of a function which indicates when the values are smaller than or equal to the corresponding quantile.

plot_quantile_ess(fit = fit_nom, par = which_min_ess, nalpha = 40)

Similar as above, we see that the efficiency of our posterior draws is worryingly low in the tails. Again, orange ticks show iterations that exceeded the maximum treedepth.

We may also investigate how the estimated effective sample sizes change when we use more and more draws (Brooks and Gelman (1998) proposed to use similar graph for \(\widehat{R}\)). If the effective sample size is highly unstable, does not increase proportionally with more draws, or even decreases, this indicates that simply running longer chains will likely not solve the convergence issues. In the plot below, we see how unstable both bulk-ESS and tail-ESS are for this example.

plot_change_ess(fit = fit_nom, par = which_min_ess)

We can further analyze potential problems using rank plots in which we clearly see differences between chains.

samp <- as.array(fit_nom)
xmin <- paste0("x[", which_min_ess, "]")
mcmc_hist_r_scale(samp[, , xmin])

4.1.2 Alternative parameterization of Cauchy

Next we examine an alternative parameterization that considers the Cauchy distribution as a scale mixture of Gaussian distributions. The model has two parameters and the Cauchy distributed \(x\)’s can be computed from those. In addition to improved sampling performance, the example illustrates that focusing on diagnostics matters.

writeLines(readLines("cauchy_alt_1.stan"))

parameters {
  vector[50] x_a;
  vector<lower=0>[50] x_b;
}

transformed parameters {
  vector[50] x = x_a ./ sqrt(x_b);
}

model {
  x_a ~ normal(0, 1);
  x_b ~ gamma(0.5, 0.5);
}

generated quantities {
  real I = fabs(x[1]) < 1 ? 1 : 0;
}

Run the alternative model:

fit_alt1 <- stan(file = 'cauchy_alt_1.stan', seed = 7878, refresh = 0)

There are no warnings, and the sampling is much faster.

mon <- monitor(fit_alt1)
print(mon)

Inference for the input samples (4 chains: each with iter = 2000; warmup = 1000):

              Q5      Q50   Q95     Mean      SD  Rhat Bulk_ESS Tail_ESS
x_a[1]   -1.6000  -0.0200   1.7 -1.5e-02    0.99  1.00     4361     2961
x_a[2]   -1.7000  -0.0016   1.6 -1.2e-02    1.00  1.00     4015     3167
x_a[3]   -1.7000  -0.0260   1.6 -9.7e-03    1.00  1.00     4190     3013
x_a[4]   -1.6000  -0.0110   1.7  6.5e-03    1.00  1.00     3626     2822
x_a[5]   -1.6000  -0.0074   1.6 -1.1e-02    0.98  1.00     3447     3029
x_a[6]   -1.6000  -0.0092   1.6 -1.1e-02    0.97  1.00     3975     2795
x_a[7]   -1.6000   0.0025   1.7  1.1e-02    1.00  1.00     3919     2271
x_a[8]   -1.7000  -0.0280   1.6 -2.4e-02    1.00  1.00     3736     3040
x_a[9]   -1.6000  -0.0370   1.5 -2.9e-02    0.99  1.00     3852     3075
x_a[10]  -1.7000  -0.0330   1.7 -4.0e-04    1.00  1.00     3457     2810
x_a[11]  -1.6000  -0.0160   1.6 -1.3e-02    0.97  1.00     3532     2822
x_a[12]  -1.6000   0.0017   1.7  8.9e-03    1.00  1.00     3462     3036
x_a[13]  -1.6000   0.0120   1.6  8.0e-03    0.99  1.00     3479     2764
x_a[14]  -1.7000  -0.0260   1.6 -2.9e-02    1.00  1.00     3872     3179
x_a[15]  -1.6000   0.0150   1.7  2.3e-02    1.00  1.00     3843     2855
x_a[16]  -1.7000  -0.0064   1.6 -1.2e-02    1.00  1.00     3610     3047
x_a[17]  -1.6000  -0.0081   1.7 -7.1e-03    0.99  1.00     4363     3076
x_a[18]  -1.7000  -0.0100   1.7 -8.0e-03    1.00  1.00     4636     3088
x_a[19]  -1.6000  -0.0200   1.6 -2.3e-02    0.98  1.00     4055     3037
x_a[20]  -1.6000  -0.0064   1.6  1.5e-04    1.00  1.00     4340     2792
x_a[21]  -1.7000   0.0400   1.7  2.5e-02    1.00  1.00     3628     2719
x_a[22]  -1.6000  -0.0130   1.6 -1.0e-02    1.00  1.00     3976     2775
x_a[23]  -1.6000   0.0340   1.7  1.8e-02    1.00  1.00     4266     2682
x_a[24]  -1.7000   0.0016   1.7 -6.5e-04    1.00  1.00     3710     2889
x_a[25]  -1.6000   0.0160   1.7  4.0e-03    1.00  1.00     4408     3062
x_a[26]  -1.6000  -0.0170   1.6 -4.9e-03    1.00  1.00     3854     2674
x_a[27]  -1.7000  -0.0010   1.7  1.5e-03    1.00  1.00     3331     2944
x_a[28]  -1.7000   0.0180   1.7  1.6e-02    1.00  1.00     4055     2689
x_a[29]  -1.6000   0.0015   1.6  1.8e-02    0.99  1.00     4097     3213
x_a[30]  -1.6000  -0.0049   1.6  1.1e-02    0.99  1.00     3961     2900
x_a[31]  -1.6000   0.0130   1.6 -4.6e-03    0.99  1.00     4097     3088
x_a[32]  -1.6000  -0.0059   1.6 -1.0e-02    0.97  1.00     4189     3171
x_a[33]  -1.7000  -0.0099   1.8  4.3e-03    1.00  1.00     3853     2594
x_a[34]  -1.7000   0.0085   1.7 -4.0e-03    1.00  1.00     4012     2787
x_a[35]  -1.6000   0.0260   1.7  2.7e-02    1.00  1.00     3920     2829
x_a[36]  -1.7000   0.0038   1.6 -7.3e-03    0.99  1.00     4107     2797
x_a[37]  -1.7000  -0.0270   1.6 -1.8e-02    1.00  1.00     3956     2942
x_a[38]  -1.7000  -0.0180   1.6 -1.6e-02    1.00  1.00     4210     2919
x_a[39]  -1.7000  -0.0250   1.6 -1.6e-02    1.00  1.00     4482     2869
x_a[40]  -1.6000   0.0028   1.6  4.3e-04    1.00  1.00     4139     2848
x_a[41]  -1.7000   0.0160   1.6 -3.2e-03    0.99  1.00     3963     2962
x_a[42]  -1.6000   0.0280   1.6  2.0e-02    1.00  1.00     4161     3077
x_a[43]  -1.6000  -0.0300   1.6 -1.5e-02    1.00  1.00     3808     2849
x_a[44]  -1.7000  -0.0330   1.6 -1.7e-02    1.00  1.00     3743     2865
x_a[45]  -1.7000   0.0190   1.7  9.2e-03    1.00  1.00     3783     2546
x_a[46]  -1.6000   0.0440   1.7  4.9e-02    0.97  1.00     4340     3277
x_a[47]  -1.7000   0.0180   1.6 -4.0e-03    1.00  1.00     4283     2946
x_a[48]  -1.6000   0.0025   1.7  2.3e-03    0.98  1.00     4001     2779
x_a[49]  -1.6000   0.0023   1.6  2.5e-03    1.00  1.00     3906     3099
x_a[50]  -1.6000  -0.0055   1.6 -3.0e-04    0.99  1.00     3794     2962
x_b[1]    0.0033   0.4400   4.0  1.0e+00    1.40  1.00     2268     1264
x_b[2]    0.0041   0.4600   3.8  1.0e+00    1.40  1.00     2444     1428
x_b[3]    0.0042   0.4700   3.9  1.0e+00    1.50  1.00     3578     1950
x_b[4]    0.0043   0.4600   3.8  1.0e+00    1.40  1.00     2693     1342
x_b[5]    0.0040   0.4500   4.0  1.0e+00    1.50  1.00     3056     1731
x_b[6]    0.0041   0.4400   3.8  1.0e+00    1.40  1.00     3264     1786
x_b[7]    0.0050   0.4300   3.6  9.5e-01    1.30  1.00     2888     1907
x_b[8]    0.0039   0.4400   3.8  1.0e+00    1.40  1.00     2876     1578
x_b[9]    0.0048   0.5000   3.7  1.0e+00    1.40  1.00     2820     1535
x_b[10]   0.0033   0.4400   3.8  9.9e-01    1.40  1.00     2534     1699
x_b[11]   0.0058   0.4900   3.9  1.0e+00    1.40  1.00     3595     2032
x_b[12]   0.0040   0.4400   3.8  9.9e-01    1.40  1.00     2405     1200
x_b[13]   0.0033   0.4600   4.0  1.0e+00    1.50  1.00     2045     1073
x_b[14]   0.0038   0.4400   4.0  1.0e+00    1.50  1.00     2829     1443
x_b[15]   0.0051   0.4500   3.8  1.0e+00    1.40  1.00     2853     1447
x_b[16]   0.0037   0.4800   3.8  1.0e+00    1.40  1.00     2661     1604
x_b[17]   0.0056   0.4600   3.9  1.0e+00    1.40  1.00     2775     1477
x_b[18]   0.0057   0.5000   4.1  1.1e+00    1.50  1.00     2689     1170
x_b[19]   0.0035   0.4500   3.9  1.0e+00    1.40  1.00     2392     1450
x_b[20]   0.0032   0.4200   4.0  9.9e-01    1.40  1.00     2296     1240
x_b[21]   0.0051   0.4900   3.9  1.0e+00    1.40  1.00     3069     1848
x_b[22]   0.0043   0.4500   3.9  1.0e+00    1.50  1.00     3012     1733
x_b[23]   0.0038   0.4600   3.9  1.0e+00    1.40  1.00     1787     1093
x_b[24]   0.0027   0.4400   3.9  1.0e+00    1.40  1.00     1903     1008
x_b[25]   0.0037   0.4500   3.7  9.8e-01    1.40  1.00     2348     1094
x_b[26]   0.0048   0.4700   4.0  1.0e+00    1.50  1.00     2421     1549
x_b[27]   0.0038   0.4500   3.9  1.0e+00    1.40  1.00     2777     1470
x_b[28]   0.0043   0.4600   3.8  9.8e-01    1.40  1.00     3353     1699
x_b[29]   0.0061   0.4600   3.9  1.0e+00    1.40  1.00     3428     1997
x_b[30]   0.0040   0.4400   3.9  1.0e+00    1.40  1.00     2833     1554
x_b[31]   0.0043   0.4900   3.8  1.0e+00    1.40  1.00     3035     1633
x_b[32]   0.0033   0.4300   3.8  9.7e-01    1.40  1.00     2276     1602
x_b[33]   0.0042   0.4500   3.8  1.0e+00    1.40  1.00     3093     1888
x_b[34]   0.0043   0.4700   4.0  1.0e+00    1.50  1.00     3309     1650
x_b[35]   0.0042   0.4800   3.8  1.0e+00    1.40  1.00     2493     1588
x_b[36]   0.0046   0.4400   3.9  9.9e-01    1.40  1.00     3108     1876
x_b[37]   0.0040   0.4600   3.7  9.8e-01    1.40  1.00     2644     1322
x_b[38]   0.0052   0.4500   3.9  1.0e+00    1.40  1.00     3155     1776
x_b[39]   0.0037   0.4500   3.8  9.9e-01    1.40  1.00     2038      934
x_b[40]   0.0034   0.4200   3.8  9.4e-01    1.30  1.00     2657     1403
x_b[41]   0.0047   0.4600   3.8  1.0e+00    1.40  1.00     2648     1370
x_b[42]   0.0024   0.4500   3.9  1.0e+00    1.40  1.00     2334     1365
x_b[43]   0.0043   0.4700   4.0  1.0e+00    1.40  1.00     2967     1797
x_b[44]   0.0044   0.4300   3.7  9.7e-01    1.40  1.00     2557     1591
x_b[45]   0.0048   0.4400   3.7  9.6e-01    1.30  1.00     2731     1785
x_b[46]   0.0044   0.4600   3.7  1.0e+00    1.40  1.00     2538     1183
x_b[47]   0.0068   0.4700   3.8  1.0e+00    1.40  1.00     3948     2071
x_b[48]   0.0064   0.4800   3.9  1.0e+00    1.40  1.00     3207     1917
x_b[49]   0.0046   0.4700   3.7  1.0e+00    1.30  1.00     2550     1533
x_b[50]   0.0033   0.4600   4.0  1.0e+00    1.50  1.00     2881     1395
x[1]     -6.5000  -0.0190   6.5  1.1e-02   35.00  1.01     3901     2122
x[2]     -6.5000  -0.0023   6.5  3.4e+00  150.00  1.00     3767     1947
x[3]     -6.3000  -0.0300   5.9 -8.3e-02   16.00  1.00     3681     2514
x[4]     -6.8000  -0.0130   5.9 -1.3e+00   51.00  1.00     3244     2159
x[5]     -6.7000  -0.0120   5.8 -3.1e-01   30.00  1.00     3355     2430
x[6]     -5.6000  -0.0130   6.5 -1.5e+02 6500.00  1.00     3802     2536
x[7]     -6.6000   0.0029   6.4 -5.8e-01   23.00  1.00     3480     2508
x[8]     -6.5000  -0.0350   6.3 -3.7e-02   21.00  1.00     3474     2374
x[9]     -5.7000  -0.0440   6.0 -9.2e-01   56.00  1.00     3493     2262
x[10]    -6.4000  -0.0420   6.4 -2.0e-01   25.00  1.00     2871     2286
x[11]    -5.6000  -0.0200   5.5  1.2e-01   12.00  1.00     3423     2587
x[12]    -7.1000   0.0018   6.2  4.1e-01   92.00  1.00     3340     2329
x[13]    -6.4000   0.0200   6.2 -5.6e+00  200.00  1.00     3332     2164
x[14]    -6.7000  -0.0400   6.6 -2.9e+00  130.00  1.00     3693     2257
x[15]    -5.7000   0.0170   6.1 -8.6e-01   30.00  1.00     3511     1969
x[16]    -7.0000  -0.0110   7.2 -4.0e-01   25.00  1.00     3614     2406
x[17]    -5.8000  -0.0130   5.4 -6.3e-02   25.00  1.00     3880     2520
x[18]    -5.4000  -0.0120   5.9 -5.5e-01  260.00  1.00     4303     2254
x[19]    -7.2000  -0.0240   5.8  9.9e+00  550.00  1.00     3505     1931
x[20]    -7.4000  -0.0084   6.7 -3.1e+00  150.00  1.00     3411     1916
x[21]    -5.9000   0.0490   5.9  2.3e-01   46.00  1.00     3419     2179
x[22]    -7.0000  -0.0190   6.2 -2.0e-01   27.00  1.00     3554     2149
x[23]    -6.0000   0.0450   6.5  2.5e+00  110.00  1.00     3456     2098
x[24]    -6.7000   0.0025   7.0 -6.7e-01  110.00  1.00     3541     1922
x[25]    -6.3000   0.0180   6.4 -2.7e+00  100.00  1.00     4036     2356
x[26]    -5.6000  -0.0210   6.3  2.0e-01   14.00  1.00     3335     2319
x[27]    -6.4000  -0.0015   6.2 -7.8e-01   53.00  1.00     3377     2400
x[28]    -5.7000   0.0230   6.5  3.1e+00  110.00  1.00     3446     2121
x[29]    -5.4000   0.0013   5.7 -1.2e-01   18.00  1.00     3918     2430
x[30]    -5.8000  -0.0054   6.5  1.0e+00   51.00  1.00     3642     2138
x[31]    -5.9000   0.0170   5.7  8.6e-01   26.00  1.00     3595     2271
x[32]    -6.5000  -0.0130   6.2  1.1e-01   51.00  1.00     4227     2731
x[33]    -7.1000  -0.0082   6.3 -4.1e-01   23.00  1.00     3807     2376
x[34]    -6.6000   0.0110   6.1  9.5e-01   48.00  1.00     4084     2358
x[35]    -5.4000   0.0300   6.1  3.2e-01   48.00  1.00     3756     2247
x[36]    -6.2000   0.0043   6.1 -5.7e-02   28.00  1.00     3600     2386
x[37]    -6.1000  -0.0280   5.3  7.0e-01   60.00  1.00     3623     2005
x[38]    -5.7000  -0.0210   5.8  2.0e-01   13.00  1.00     3820     2536
x[39]    -6.7000  -0.0340   5.8  2.8e+00  290.00  1.00     4121     1944
x[40]    -6.4000   0.0056   6.5 -2.7e+00  100.00  1.00     3612     1823
x[41]    -6.0000   0.0220   5.9 -4.2e-01   36.00  1.00     3492     2222
x[42]    -7.4000   0.0450   6.9  4.9e-01   22.00  1.00     3558     1949
x[43]    -6.0000  -0.0340   6.7  1.4e+01  630.00  1.00     3823     2516
x[44]    -7.0000  -0.0390   6.2  1.5e+00  110.00  1.00     3310     2239
x[45]    -5.7000   0.0210   6.2 -4.3e-01   32.00  1.00     3752     2437
x[46]    -5.6000   0.0600   6.3 -3.2e-01   76.00  1.00     3898     1976
x[47]    -5.5000   0.0290   5.4 -2.1e-02   12.00  1.00     3893     2659
x[48]    -6.0000   0.0032   4.8 -1.3e-02   21.00  1.00     3674     2274
x[49]    -6.5000   0.0028   5.2 -1.2e+00  130.00  1.00     3576     2243
x[50]    -6.7000  -0.0078   6.9 -1.1e+00  150.00  1.00     3437     2486
I         0.0000   0.0000   1.0  5.0e-01    0.50  1.00     2648     4000
lp__    -95.0000 -81.0000 -69.0 -8.1e+01    8.10  1.00     1310     1928

For each parameter, Bulk_ESS and Tail_ESS are crude measures of 
effective sample size for bulk and tail quantities respectively (good values is 
ESS > 400), and Rhat is the potential scale reduction factor on rank normalized
split chains (at convergence, Rhat = 1).

which_min_ess <- which.min(mon[101:150, 'Tail_ESS'])

All Rhat < 1.01 and ESS > 400 indicate the sampling worked much better with the alternative parameterization. Appendix E has more results using other alternative parameterizations. The x_a and x_b used to form the Cauchy distributed x have stable quantile, mean and sd values. As x is Cauchy distributed it has only stable quantiles, but wildly varying mean and sd estimates as the true values are not finite.

We can further analyze potential problems using local efficiency estimates and rank plots. We take a detailed look at x[40], which has the smallest bulk-ESS of 2848.

We examine the sampling efficiency in different parts of the posterior by computing the efficiency estimates for small interval probability estimates.

plot_local_ess(fit = fit_alt1, par = which_min_ess + 100, nalpha = 20)

The efficiency estimate is good in all parts of the posterior. Further, we examine the sampling efficiency of different quantile estimates.

plot_quantile_ess(fit = fit_alt1, par = which_min_ess + 100, nalpha = 40)

Rank plots also look rather similar across chains.

samp <- as.array(fit_alt1)
xmin <- paste0("x[", which_min_ess, "]")
mcmc_hist_r_scale(samp[, , xmin])

In summary, the alternative parameterization produces results that look much better than for the nominal parameterization. There are still some differences in the tails, which we also identified via the tail-ESS.

4.1.3 Half-Cauchy with nominal parameterization

Half-Cauchy priors are common and, for example, in Stan usually set using the nominal parameterization. However, when the constraint <lower=0> is used, Stan does the sampling automatically in the unconstrained log(x) space, which changes the geometry crucially.

writeLines(readLines("half_cauchy_nom.stan"))

parameters {
  vector<lower=0>[50] x;
}

model {
  x ~ cauchy(0, 1);
}

generated quantities {
  real I = fabs(x[1]) < 1 ? 1 : 0;
}

Run the half-Cauchy with nominal parameterization (and positive constraint):

fit_half_nom <- stan(file = 'half_cauchy_nom.stan', seed = 7878, refresh = 0)

There are no warnings, and the sampling is much faster than for the Cauchy nominal model.

mon <- monitor(fit_half_nom)
print(mon)

Inference for the input samples (4 chains: each with iter = 2000; warmup = 1000):

           Q5    Q50 Q95   Mean     SD  Rhat Bulk_ESS Tail_ESS
x[1]    0.081   1.00  14  11.00  390.0 1.000     8077     2223
x[2]    0.096   1.00  11  12.00  480.0 1.000     9868     2612
x[3]    0.063   1.00  13   5.40   72.0 1.000     7895     2097
x[4]    0.094   0.98  12   4.40   29.0 1.000     7596     2347
x[5]    0.076   1.00  14   4.50   20.0 1.000     7495     2230
x[6]    0.081   0.99  12  13.00  380.0 1.000     7948     2145
x[7]    0.073   0.98  12   9.20  280.0 1.000     8336     2117
x[8]    0.090   0.99  11   4.60   41.0 1.000     8194     2165
x[9]    0.090   1.00  12  12.00  480.0 1.000     8464     2127
x[10]   0.072   1.00  15   6.00   51.0 1.000     7963     2399
x[11]   0.075   0.97  13  16.00  530.0 1.000     6980     1788
x[12]   0.073   1.00  13   4.10   17.0 1.000     8226     2029
x[13]   0.077   1.00  14   8.90  120.0 1.000     8077     2032
x[14]   0.068   0.96  13   6.90   72.0 1.000     7455     2261
x[15]   0.070   1.00  14   8.70   96.0 1.000     6766     2246
x[16]   0.075   0.99  14   5.60   38.0 1.000     7396     2174
x[17]   0.082   0.99  12   8.70  300.0 1.000     7144     2260
x[18]   0.085   0.98  12  14.00  400.0 1.000     7614     2035
x[19]   0.073   0.99  14   6.30   69.0 1.000     7823     1825
x[20]   0.090   1.00  11   6.40  160.0 1.000     7905     2355
x[21]   0.073   1.00  12   7.00  140.0 0.999     7843     2078
x[22]   0.087   1.00  13  32.00 1700.0 1.000     7735     2043
x[23]   0.070   0.98  13   6.30   67.0 1.000     7119     2142
x[24]   0.073   1.00  12   5.00   42.0 1.000     6893     1982
x[25]   0.090   1.00  12   8.60  270.0 1.000     7757     2351
x[26]   0.085   0.98  11   7.00  120.0 1.000     6433     2230
x[27]   0.077   1.00  13   4.90   34.0 1.000     7005     2119
x[28]   0.081   0.97  11   5.80   67.0 1.000     9631     2228
x[29]   0.074   1.00  14  10.00  240.0 1.000     6109     2312
x[30]   0.089   1.00  11   8.00  210.0 1.000     7958     2368
x[31]   0.078   0.96  12   4.40   32.0 1.000     6493     2102
x[32]   0.097   1.00  11   5.00   55.0 1.000     7043     1742
x[33]   0.076   0.99  14   5.50   37.0 1.000     6913     2455
x[34]   0.077   0.98  12   5.90   78.0 1.000     8610     2514
x[35]   0.075   0.96  14   5.70   57.0 1.000     6406     2160
x[36]   0.064   1.00  14   5.40   40.0 1.000     7694     2031
x[37]   0.071   1.00  14   8.30  200.0 1.000     7276     2491
x[38]   0.083   1.00  12   6.40  120.0 1.000     6790     2369
x[39]   0.095   1.00  12   6.70   89.0 1.000     7739     2518
x[40]   0.092   0.96  12   5.20   44.0 1.000     7087     2349
x[41]   0.070   0.96  13   5.00   42.0 1.000     8650     2333
x[42]   0.071   1.00  13   7.80  130.0 1.000     8703     2410
x[43]   0.076   0.97  10  27.00 1400.0 1.000     8747     2194
x[44]   0.079   1.00  12   6.00   59.0 1.000     6378     2257
x[45]   0.080   0.98  12   7.70  120.0 1.000     8430     2314
x[46]   0.082   0.99  12   5.40   72.0 1.000     8185     2237
x[47]   0.084   1.00  14   7.00   77.0 1.000     9562     2265
x[48]   0.076   1.00  13   5.10   30.0 1.000     8402     2690
x[49]   0.079   1.00  13   7.50  100.0 1.000     7993     1804
x[50]   0.084   1.00  13   8.90  180.0 1.000     7523     2243
I       0.000   0.00   1   0.49    0.5 0.999     7357     4000
lp__  -81.000 -69.00 -59 -69.00    6.4 1.000     1218     2001

For each parameter, Bulk_ESS and Tail_ESS are crude measures of 
effective sample size for bulk and tail quantities respectively (good values is 
ESS > 400), and Rhat is the potential scale reduction factor on rank normalized
split chains (at convergence, Rhat = 1).

All Rhat < 1.01 and ESS > 400 indicate good performance of the sampler. We see that the Stan’s automatic (and implicit) transformation of constraint parameters can have a big effect on the sampling performance. More experiments with different parameterizations of the half-Cauchy distribution can be found in Appendix E.

4.2 Hierarchical model: Eight Schools

The Eight Schools data is a classic example for hierarchical models (see Section 5.5 in Gelman et al., 2013), which despite the apparent simplicity nicely illustrates the typical problems in inference for hierarchical models. The Stan models below are from Michael Betancourt’s case study on Diagnosing Biased Inference with Divergences. Appendix F contains more detailed analysis of different algorithm variants.

4.2.1 A Centered Eight Schools model

writeLines(readLines("eight_schools_cp.stan"))

data {
  int<lower=0> J;
  real y[J];
  real<lower=0> sigma[J];
}

parameters {
  real mu;
  real<lower=0> tau;
  real theta[J];
}

model {
  mu ~ normal(0, 5);
  tau ~ cauchy(0, 5);
  theta ~ normal(mu, tau);
  y ~ normal(theta, sigma);
}

4.2.1.1 Centered Eight Schools model

We directly run the centered parameterization model with an increased adapt_delta value to reduce the probability of getting divergent transitions.

eight_schools <- read_rdump("eight_schools.data.R")
fit_cp <- stan(
  file = 'eight_schools_cp.stan', data = eight_schools,
  iter = 2000, chains = 4, seed = 483892929, refresh = 0,
  control = list(adapt_delta = 0.95)
)

Warning: There were 113 divergent transitions after warmup. Increasing adapt_delta above 0.95 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup

Warning: There were 2 chains where the estimated Bayesian Fraction of Missing Information was low. See
http://mc-stan.org/misc/warnings.html#bfmi-low

Warning: Examine the pairs() plot to diagnose sampling problems

Despite an increased adapt_delta, we still observe a lot of divergent transitions, which in itself is already sufficient indicator to not trust the results. We can use Rhat and ESS diagnostics to recognize problematic parts of the posterior and they could be used in cases when other MCMC algorithms than HMC is used.

mon <- monitor(fit_cp)
print(mon)

Inference for the input samples (4 chains: each with iter = 2000; warmup = 1000):

             Q5   Q50   Q95  Mean  SD  Rhat Bulk_ESS Tail_ESS
mu        -1.10   4.5  9.90   4.4 3.4  1.02      548      754
tau        0.39   2.8  9.60   3.6 3.1  1.07       67       82
theta[1]  -2.20   5.8 16.00   6.2 5.7  1.02      747     1294
theta[2]  -2.60   5.1 13.00   5.1 4.9  1.01      970     1240
theta[3]  -5.00   4.4 12.00   3.9 5.3  1.01      899     1147
theta[4]  -2.90   5.0 13.00   4.9 4.8  1.01      986     1059
theta[5]  -4.70   4.0 11.00   3.7 4.8  1.01      715      988
theta[6]  -4.20   4.3 12.00   4.1 4.8  1.01      833      976
theta[7]  -1.30   6.0 16.00   6.3 5.2  1.02      612     1182
theta[8]  -3.40   5.1 14.00   5.0 5.3  1.01      901     1477
lp__     -25.00 -15.0  0.37 -14.0 7.4  1.07       69       89

For each parameter, Bulk_ESS and Tail_ESS are crude measures of 
effective sample size for bulk and tail quantities respectively (good values is 
ESS > 400), and Rhat is the potential scale reduction factor on rank normalized
split chains (at convergence, Rhat = 1).

See Appendix F for results of longer chains.

Bulk-ESS and Tail-ESS for the between school standard deviation tau are 67 and 82 respectively. Both are less than 400, indicating we should investigate that parameter more carefully. We thus examine the sampling efficiency in different parts of the posterior by computing the efficiency estimate for small interval estimates for tau. These plots may either show quantiles or parameter values at the vertical axis. Red ticks show divergent transitions.

plot_local_ess(fit = fit_cp, par = "tau", nalpha = 20)

plot_local_ess(fit = fit_cp, par = "tau", nalpha = 20, rank = FALSE)

We see that the sampler has difficulties in exploring small tau values. As the sampling efficiency for estimating small tau values is practically zero, we may assume that we may miss substantial amount of posterior mass and get biased estimates. Red ticks, which show iterations with divergences, have concentrated to small tau values, indicate also problems exploring small values which is likely to cause bias.

We examine also the sampling efficiency of different quantile estimates. Again, these plots may either show quantiles or parameter values at the vertical axis.

plot_quantile_ess(fit = fit_cp, par = "tau", nalpha = 40)

plot_quantile_ess(fit = fit_cp, par = "tau", nalpha = 40, rank = FALSE)

Most of the quantile estimates have worryingly low effective sample size.

Let’s see how the estimated effective sample size changes when we use more and more draws. Here we don’t see sudden changes, but both bulk-ESS and tail-ESS are too low. See Appendix F for results of longer chains.

plot_change_ess(fit = fit_cp, par = "tau")

In lines with these findings, the rank plots of tau clearly show problems in the mixing of the chains.

samp_cp <- as.array(fit_cp)
mcmc_hist_r_scale(samp_cp[, , "tau"])

4.2.2 Non-centered Eight Schools model

For hierarchical models, the non-centered parameterization often works better than the centered one:

writeLines(readLines("eight_schools_ncp.stan"))

data {
  int<lower=0> J;
  real y[J];
  real<lower=0> sigma[J];
}

parameters {
  real mu;
  real<lower=0> tau;
  real theta_tilde[J];
}

transformed parameters {
  real theta[J];
  for (j in 1:J)
    theta[j] = mu + tau * theta_tilde[j];
}

model {
  mu ~ normal(0, 5);
  tau ~ cauchy(0, 5);
  theta_tilde ~ normal(0, 1);
  y ~ normal(theta, sigma);
}

For reasons of comparability, we also run the non-centered parameterization model with an increased adapt_delta value:

fit_ncp2 <- stan(
  file = 'eight_schools_ncp.stan', data = eight_schools,
  iter = 2000, chains = 4, seed = 483892929, refresh = 0,
  control = list(adapt_delta = 0.95)
)

We get zero divergences and no other warnings which is a first good sign.

mon <- monitor(fit_ncp2)
print(mon)

Inference for the input samples (4 chains: each with iter = 2000; warmup = 1000):

                   Q5    Q50  Q95   Mean   SD  Rhat Bulk_ESS Tail_ESS
mu              -1.10  4.400 10.0  4.500 3.40     1     5531     3004
tau              0.30  2.800  9.5  3.600 3.20     1     2872     1908
theta_tilde[1]  -1.30  0.310  1.9  0.310 0.98     1     5046     2874
theta_tilde[2]  -1.40  0.110  1.6  0.100 0.94     1     4177     2735
theta_tilde[3]  -1.60 -0.075  1.5 -0.078 0.97     1     6485     2994
theta_tilde[4]  -1.50  0.084  1.6  0.064 0.95     1     6076     2514
theta_tilde[5]  -1.70 -0.170  1.4 -0.160 0.94     1     5608     3177
theta_tilde[6]  -1.60 -0.060  1.5 -0.065 0.97     1     4855     2773
theta_tilde[7]  -1.30  0.400  1.9  0.370 0.96     1     4796     2849
theta_tilde[8]  -1.50  0.072  1.7  0.062 0.96     1     6142     2972
theta[1]        -1.60  5.600 16.0  6.200 5.60     1     4907     3015
theta[2]        -2.40  4.800 13.0  5.000 4.70     1     5122     3242
theta[3]        -4.60  4.300 12.0  4.100 5.30     1     5457     3407
theta[4]        -2.70  4.800 12.0  4.800 4.80     1     4695     3130
theta[5]        -4.00  3.800 11.0  3.700 4.60     1     5346     3398
theta[6]        -3.90  4.300 11.0  4.100 5.00     1     5670     3393
theta[7]        -0.98  5.900 15.0  6.400 5.10     1     4708     3242
theta[8]        -3.20  4.800 13.0  4.900 5.20     1     4924     3037
lp__           -11.00 -6.600 -3.8 -6.900 2.30     1     1641     2344

For each parameter, Bulk_ESS and Tail_ESS are crude measures of 
effective sample size for bulk and tail quantities respectively (good values is 
ESS > 400), and Rhat is the potential scale reduction factor on rank normalized
split chains (at convergence, Rhat = 1).

All Rhat < 1.01 and ESS > 400 indicate a much better performance of the non-centered parameterization.

We examine the sampling efficiency in different parts of the posterior by computing the effective sample size for small interval probability estimates for tau.

plot_local_ess(fit = fit_ncp2, par = 2, nalpha = 20)

Small tau values are still more difficult to explore, but the relative efficiency is in a good range. We may also check this with a finer resolution:

plot_local_ess(fit = fit_ncp2, par = 2, nalpha = 50)

The sampling efficiency for different quantile estimates looks good as well.

plot_quantile_ess(fit = fit_ncp2, par = 2, nalpha = 40)

In line with these findings, the rank plots of tau show no substantial differences between chains.

samp_ncp2 <- as.array(fit_ncp2)
mcmc_hist_r_scale(samp_ncp2[, , 2])

References

Betancourt, M. (2017) ‘A conceptual introduction to hamiltonian monte carlo’, arXiv preprint arXiv:1701.02434.

Brooks, S. P. and Gelman, A. (1998) ‘General methods for monitoring convergence of iterative simulations’, Journal of Computational and Graphical Statistics, 7(4), pp. 434–455.

Gelman, A. et al. (2013) Bayesian data analysis, third edition. CRC Press.

Gelman, A. and Rubin, D. B. (1992) ‘Inference from iterative simulation using multiple sequences’, Statistical science, 7(4), pp. 457–472.

Geyer, C. J. (1992) ‘Practical Markov chain Monte Carlo’, Statistical Science, 7, pp. 473–483.

Geyer, C. J. (2011) ‘Introduction to Markov chain Monte Carlo’, in Brooks, S. et al. (eds) Handbook of markov chain monte carlo. CRC Press.

Hoffman, M. D. and Gelman, A. (2014) ‘The No-U-Turn Sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo’, Journal of Machine Learning Research, 15, pp. 1593–1623. Available at: http://jmlr.org/papers/v15/hoffman14a.html.

Stan Development Team (2018a) Bayesian statistics using stan. Stan Development Team. Available at: https://github.com/stan-dev/stan-book.

Stan Development Team (2018b) ‘RStanArm: Bayesian applied regression modeling via Stan. R package version 2.17.4’. Available at: http://mc-stan.org.

Stan Development Team (2018c) ‘RStan: The R interface to Stan. R package version 2.17.3’. Available at: http://mc-stan.org.

Stan Development Team (2018d) ‘Stan modeling language users guide and reference manual. Version 2.18.0’. Available at: http://mc-stan.org.

Stan Development Team (2018e) ‘The Stan core library version 2.18.0’. Available at: http://mc-stan.org.

Appendices

Appendix A: Abbreviations

The following abbreviations are used throughout the appendices:

N = total number of draws
Rhat = classic no-split-Rhat
sRhat = classic split-Rhat
zsRhat = rank-normalized split-Rhat
- all chains are jointly ranked and z-transformed
- can detect differences in location and trends
zfsRhat = rank-normalized folded split-Rhat
- all chains are jointly “folded” by computing absolute deviation from median, ranked and z-transformed
- can detect differences in scales
seff = no-split effective sample size
reff = seff / N
zsseff = rank-normalized split effective sample size
- estimates the efficiency of mean estimate for rank normalized values
zsreff = zsseff / N
zfsseff = rank-normalized folded split effective sample size
- estimates the efficiency of rank normalized mean absolute deviation
zfsreff = zfsseff / N
tailseff = minimum of rank-normalized split effective sample sizes of the 5% and 95% quantiles
tailreff = tailseff / N
medsseff = median split effective sample size
- estimates the efficiency of the median
medsreff = medsseff / N
madsseff = mad split effective sample size
- estimates the efficiency of the median absolute deviation
madsreff = madsseff / N

Appendix B: Examples of rank normalization

We will illustrate the rank normalization with a few examples. First, we plot histograms, and empirical cumulative distribution functions (ECDF) with respect to the original parameter values (\(\theta\)), scaled ranks (ranks divided by the maximum rank), and rank normalized values (z). We used scaled ranks to make the plots look similar for different number of draws.

100 draws from Normal(0, 1):

n <- 100
theta <- rnorm(n)
plot_ranknorm(theta, n)

100 draws from Exponential(1):

theta <- rexp(n)
plot_ranknorm(theta, n)

100 draws from Cauchy(0, 1):

theta <- rcauchy(n)
plot_ranknorm(theta, n)

In the above plots, the ECDF with respect to scaled rank and rank normalized \(z\)-values look exactly the same for all distributions. In Split-\(\widehat{R}\) and effective sample size computations, we rank all draws jointly, but then compare ranks and ECDF of individual split chains. To illustrate the variation between chains, we draw 8 batches of 100 draws each from Normal(0, 1):

n <- 100
m <- 8
theta <- rnorm(n * m)
plot_ranknorm(theta, n, m)

The variation in ECDF due to the variation ranks is now visible also in scaled ranks and rank normalized \(z\)-values from different batches (chains).

The benefit of rank normalization is more obvious for non-normal distribution such as Cauchy:

theta <- rcauchy(n * m)
plot_ranknorm(theta, n, m)

Rank normalization makes the subsequent computations well defined and invariant under bijective transformations. This means that we get the same results, for example, if we use unconstrained or constrained parameterisations in a model.

Appendix C: Variance of the cumulative distribution function

In Section 3, we had defined the empirical CDF (ECDF) for any \(\theta_\alpha\) as \[ p(\theta \leq \theta_\alpha) \approx \bar{I}_\alpha = \frac{1}{S}\sum_{s=1}^S I(\theta^{(s)} \leq\theta_\alpha), \]

For independent draws, \(\bar{I}_\alpha\) has a \({\rm Beta}(\bar{I}_\alpha+1, S - \bar{I}_\alpha + 1)\) distribution. Thus we can easily examine the variation of the ECDF for any \(\theta_\alpha\) value from a single chain. If \(\bar{I}_\alpha\) is not very close to \(1\) or \(S\) and \(S\) is large, we can use the variance of Beta distribution

\[ {\rm Var}[p(\theta \leq \theta_\alpha)] = \frac{(\bar{I}_\alpha+1)*(S-\bar{I}_\alpha+1)}{(S+2)^2(S+3)}. \] We illustrate uncertainty intervals of the Beta distribution and normal approximation of ECDF for 100 draws from Normal(0, 1):

n <- 100
m <- 1
theta <- rnorm(n * m)
plot_ranknorm(theta, n, m, interval = TRUE)

Uncertainty intervals of ECDF for draws from Cauchy(0, 1) illustrate again the improved visual clarity in plotting when using scaled ranks:

n <- 100
m <- 1
theta <- rcauchy(n * m)
plot_ranknorm(theta, n, m, interval = TRUE)

The above plots illustrate that the normal approximation is accurate for practical purposes in MCMC diagnostics.

Appendix D: Normal distributions with additional trend, shift or scaling

This part focuses on diagnostics for

all chains having a trend and a similar marginal distribution
one of the chains having a different mean
one of the chains having a lower marginal variance

To simplify, in this part, independent draws are used as a proxy for very efficient MCMC sampling. First, we sample draws from a standard-normal distribution. We will discuss the behavior for non-normal distributions later. See Appendix A for the abbreviations used.

Adding the same trend to all chains

All chains are from the same Normal(0, 1) distribution plus a linear trend added to all chains:

conds <- expand.grid(
  iters = c(250, 1000, 4000), 
  trend = c(0, 0.25, 0.5, 0.75, 1),
  rep = 1:10
)
res <- vector("list", nrow(conds))
chains = 4
for (i in 1:nrow(conds)) {
  iters <- conds[i, "iters"]
  trend <- conds[i, "trend"]
  rep <- conds[i, "rep"]
  r <- array(rnorm(iters * chains), c(iters, chains))
  r <- r + seq(-trend, trend, length.out = iters)
  rs <- as.data.frame(monitor_extra(r))
  res[[i]] <- cbind(iters, trend, rep, rs)
}
res <- bind_rows(res)

If we don’t split chains, Rhat misses the trends if all chains still have a similar marginal distribution.

ggplot(data = res, aes(y = Rhat, x = trend)) + 
  geom_point() + 
  geom_jitter() + 
  facet_grid(. ~ iters) + 
  geom_hline(yintercept = 1.005, linetype = 'dashed') + 
  geom_hline(yintercept = 1) + 
  ggtitle('Rhat without splitting chains')

Split-Rhat can detect trends, even if the marginals of the chains are similar.

ggplot(data = res, aes(y = zsRhat, x = trend)) + 
  geom_point() + geom_jitter() + 
  facet_grid(. ~ iters) + 
  geom_hline(yintercept = 1.005, linetype = 'dashed') + 
  geom_hline(yintercept = 1) + 
  ggtitle('Split-Rhat')

Result: Split-Rhat is useful for detecting non-stationarity (i.e., trends) in the chains. If we use a threshold of \(1.01\), we can detect trends which account for 2% or more of the total marginal variance. If we use a threshold of \(1.1\), we detect trends which account for 30% or more of the total marginal variance.

The effective sample size is based on split Rhat and within-chain autocorrelation. We plot the relative efficiency \(R_{\rm eff}=S_{\rm eff}/S\) for easier comparison between different values of \(S\). In the plot below, dashed lines indicate the threshold at which we would consider the effective sample size to be sufficient (i.e., \(S_{\rm eff} > 400\)). Since we plot the relative efficiency instead of the effective sample size itself, this threshold is divided by \(S\), which we compute here as the number of iterations per chain (variable iter) times the number of chains (\(4\)).

ggplot(data = res, aes(y = zsreff, x = trend)) + 
  geom_point() + 
  geom_jitter() + 
  facet_grid(. ~ iters) + 
  geom_hline(yintercept = c(0,1)) + 
  geom_hline(aes(yintercept = 400 / (4 * iters)), linetype = 'dashed') + 
  ggtitle('Relative Bulk-ESS (zsreff)') + 
  scale_y_continuous(breaks = seq(0, 1.5, by = 0.25))

Result: Split-Rhat is more sensitive to trends for small sample sizes, but effective sample size becomes more sensitive for larger samples sizes (as autocorrelations can be estimated more accurately).

Advice: If in doubt, run longer chains for more accurate convergence diagnostics.

Shifting one chain

Next we investigate the sensitivity to detect if one of the chains has not converged to the same distribution as the others, but has a different mean.

conds <- expand.grid(
  iters = c(250, 1000, 4000), 
  shift = c(0, 0.25, 0.5, 0.75, 1),
  rep = 1:10
)
res <- vector("list", nrow(conds))
chains = 4
for (i in 1:nrow(conds)) {
  iters <- conds[i, "iters"]
  shift <- conds[i, "shift"]
  rep <- conds[i, "rep"]
  r <- array(rnorm(iters * chains), c(iters, chains))
  r[, 1] <- r[, 1] + shift
  rs <- as.data.frame(monitor_extra(r))
  res[[i]] <- cbind(iters, shift, rep, rs)
}
res <- bind_rows(res)

ggplot(data = res, aes(y = zsRhat, x = shift)) + 
  geom_point() + 
  geom_jitter() + 
  facet_grid(. ~ iters) + 
  geom_hline(yintercept = 1.005, linetype = 'dashed') + 
  geom_hline(yintercept = 1) + 
  ggtitle('Split-Rhat')

Result: If we use a threshold of \(1.01\), we can detect shifts with a magnitude of one third or more of the marginal standard deviation. If we use a threshold of \(1.1\), we detect shifts with a magnitude equal to or larger than the marginal standard deviation.

ggplot(data = res, aes(y = zsreff, x = shift)) + 
  geom_point() +
  geom_jitter() + 
  facet_grid(. ~ iters) + 
  geom_hline(yintercept = c(0,1)) + 
  geom_hline(aes(yintercept = 400 / (4 * iters)), linetype = 'dashed') + 
  ggtitle('Relative Bulk-ESS (zsreff)') + 
  scale_y_continuous(breaks = seq(0, 1.5, by = 0.25))

Result: The effective sample size is not as sensitive, but a shift with a magnitude of half the marginal standard deviation or more will lead to very low relative efficiency when the total number of draws increases.

Rank plots can be used to visualize differences between chains. Here, we show rank plots for the case of 4 chains, 250 draws per chain, and a shift of 0.5.

iters = 250
chains = 4
shift = 0.5
r <- array(rnorm(iters * chains), c(iters, chains))
r[, 1] <- r[, 1] + shift
colnames(r) <- 1:4
mcmc_hist_r_scale(r)

Although, Rhat was less than \(1.05\) for this situation, the rank plots clearly show that the first chains behaves differently.

Scaling one chain

Next, we investigate the sensitivity to detect if one of the chains has not converged to the same distribution as the others, but has lower marginal variance.

conds <- expand.grid(
  iters = c(250, 1000, 4000), 
  scale = c(0, 0.25, 0.5, 0.75, 1),
  rep = 1:10
)
res <- vector("list", nrow(conds))
chains = 4
for (i in 1:nrow(conds)) {
  iters <- conds[i, "iters"]
  scale <- conds[i, "scale"]
  rep <- conds[i, "rep"]
  r <- array(rnorm(iters * chains), c(iters, chains))
  r[, 1] <- r[, 1] * scale
  rs <- as.data.frame(monitor_extra(r))
  res[[i]] <- cbind(iters, scale, rep, rs)
}
res <- bind_rows(res)

We first look at the Rhat estimates:

ggplot(data = res, aes(y = zsRhat, x = scale)) + 
  geom_point() + 
  geom_jitter() + 
  facet_grid(. ~ iters) + 
  geom_hline(yintercept = 1.005, linetype = 'dashed') + 
  geom_hline(yintercept = 1) + 
  ggtitle('Split-Rhat')

Result: Split-Rhat is not able to detect scale differences between chains.

ggplot(data = res, aes(y = zfsRhat, x = scale)) + 
  geom_point() + 
  geom_jitter() + 
  facet_grid(. ~ iters) + 
  geom_hline(yintercept = 1.005, linetype = 'dashed') + 
  geom_hline(yintercept = 1) + 
  ggtitle('Folded-split-Rhat')

Result: Folded-Split-Rhat focuses on scales and detects scale differences.

Result: If we use a threshold of \(1.01\), we can detect a chain with scale less than \(3/4\) of the standard deviation of the others. If we use threshold of \(1.1\), we detect a chain with standard deviation less than \(1/4\) of the standard deviation of the others.

Next, we look at the effective sample size estimates:

ggplot(data = res, aes(y = zsreff, x = scale)) + 
  geom_point() + 
  geom_jitter() + 
  facet_grid(. ~ iters) + 
  geom_hline(yintercept = c(0,1)) + 
  geom_hline(aes(yintercept = 400 / (4 * iters)), linetype = 'dashed') + 
  ggtitle('Relative Bulk-ESS (zsreff)') + 
  scale_y_continuous(breaks = seq(0, 1.5, by = 0.25))

Result: The bulk effective sample size of the mean does not see a problem as it focuses on location differences between chains.

Rank plots can be used to visualize differences between chains. Here, we show rank plots for the case of 4 chains, 250 draws per chain, and with one chain having a standard deviation of 0.75 as opposed to a standard deviation of 1 for the other chains.

iters = 250
chains = 4
scale = 0.75
r <- array(rnorm(iters * chains), c(iters, chains))
r[, 1] <- r[, 1] * scale
colnames(r) <- 1:4
mcmc_hist_r_scale(r)

Although folded Rhat is \(1.06\), the rank plots clearly show that the first chains behaves differently.

Appendix E: Cauchy: A distribution with infinite mean and variance

The classic split-Rhat is based on calculating within and between chain variances. If the marginal distribution of a chain is such that the variance is not defined (i.e. infinite), the classic split-Rhat is not well justified. In this section, we will use the Cauchy distribution as an example of such distribution. Also in cases where mean and variance are finite, the distribution can be far from Gaussian. Especially distributions with very long tails cause instability for variance and autocorrelation estimates. To alleviate these problems we will use Split-Rhat for rank-normalized draws.

The following Cauchy models are from Michael Betancourt’s case study Fitting The Cauchy Distribution

Nominal parameterization of Cauchy

We already looked at the nominal Cauchy model with direct parameterization in the main text, but for completeness, we take a closer look using different variants of the diagnostics.

writeLines(readLines("cauchy_nom.stan"))

parameters {
  vector[50] x;
}

model {
  x ~ cauchy(0, 1);
}

generated quantities {
  real I = fabs(x[1]) < 1 ? 1 : 0;
}

Default Stan options

Run the nominal model:

fit_nom <- stan(file = 'cauchy_nom.stan', seed = 7878, refresh = 0)

Warning: There were 1233 transitions after warmup that exceeded the maximum treedepth. Increase max_treedepth above 10. See
http://mc-stan.org/misc/warnings.html#maximum-treedepth-exceeded

Warning: Examine the pairs() plot to diagnose sampling problems

Treedepth exceedence and Bayesian Fraction of Missing Information are dynamic HMC specific diagnostics (Betancourt, 2017). We get warnings about very large number of transitions after warmup that exceeded the maximum treedepth, which is likely due to very long tails of the Cauchy distribution. All chains have low estimated Bayesian fraction of missing information also indicating slow mixing.

Trace plots for the first parameter look wild with occasional large values:

samp <- as.array(fit_nom) 
mcmc_trace(samp[, , 1])

Let’s check Rhat and ESS diagnostics.

res <- monitor_extra(samp[, , 1:50])
which_min_ess <- which.min(res$tailseff)
plot_rhat(res)

For one parameter, Rhats exceed the classic threshold of 1.1. Depending on the Rhat estimate, a few others also exceed the threshold of 1.01. The rank normalized split-Rhat has several values over 1.01. Please note that the classic split-Rhat is not well defined in this example, because mean and variance of the Cauchy distribution are not finite.

plot_ess(res)

Both classic and new effective sample size estimates have several very small values, and so the overall sample shouldn’t be trusted.

Result: Effective sample size is more sensitive than (rank-normalized) split-Rhat to detect problems of slow mixing.

We also check the log posterior value lp__ and find out that the effective sample size is worryingly low.

res <- monitor_extra(samp[, , 51:52]) 
cat('lp__: Bulk-ESS = ', round(res['lp__', 'zsseff'], 2), '\n')

lp__: Bulk-ESS =  117

cat('lp__: Tail-ESS = ', round(res['lp__', 'tailseff'], 2), '\n')

lp__: Tail-ESS =  323

We can further analyze potential problems using local effective sample size and rank plots. We examine x[36], which has the smallest tail-ESS of 117.

We examine the sampling efficiency in different parts of the posterior by computing the effective sample size for small interval probability estimates (see Section Efficiency for small interval probability estimates). Each interval contains \(1/k\) of the draws (e.g., with \(k=20\)). The small interval efficiency measures mixing of an indicator function which indicates when the values are inside the specific small interval. This gives us a local efficiency measure which does not depend on the shape of the distribution.

plot_local_ess(fit = fit_nom, par = which_min_ess, nalpha = 20)

We see that the efficiency is worryingly low in the tails (which is caused by slow mixing in long tails of Cauchy). Orange ticks show draws that exceeded the maximum treedepth.

An alternative way to examine the effective sample size in different parts of the posterior is to compute effective sample size for quantiles (see Section Efficiency for quantiles). Each interval has a specified proportion of draws, and the efficiency measures mixing of an indicator function’s results which indicate when the values are inside the specific interval.

plot_quantile_ess(fit = fit_nom, par = which_min_ess, nalpha = 40)

We see that the efficiency is worryingly low in the tails (which is caused by slow mixing in long tails of Cauchy). Orange ticks show draws that exceeded the maximum treedepth.

We can further analyze potential problems using rank plots, from which we clearly see differences between chains.

xmin <- paste0("x[", which_min_ess, "]")
mcmc_hist_r_scale(samp[, , xmin])

Default Stan options + increased maximum treedepth

We can try to improve the performance by increasing max_treedepth to \(20\):

fit_nom_td20 <- stan(
  file = 'cauchy_nom.stan', seed = 7878, 
  refresh = 0, control = list(max_treedepth = 20)
)

Trace plots for the first parameter still look wild with occasional large values.

samp <- as.array(fit_nom_td20)
mcmc_trace(samp[, , 1])

res <- monitor_extra(samp[, , 1:50])
which_min_ess <- which.min(res$tailseff)

We check the diagnostics for all \(x\).

plot_rhat(res)

All Rhats are below \(1.1\), but many are over \(1.01\). Classic split-Rhat has more variation than the rank normalized Rhat (note that the former is not well defined). The folded rank normalized Rhat shows that there is still more variation in the scale than in the location between different chains.

plot_ess(res)

Some classic effective sample sizes are very small. If we wouldn’t realize that the variance is infinite, we might try to run longer chains, but in case of an infinite variance, zero relative efficiency (ESS/S) is the truth and longer chains won’t help with that. However other quantities can be well defined, and that’s why it is useful to also look at the rank normalized version as a generic transformation to achieve finite mean and variance. The smallest bulk-ESS is less than 1000, which is not that bad. The smallest median-ESS is larger than 2500, that is we are able to estimate the median efficiently. However, many tail-ESS’s are less than 400 indicating problems for estimating the scale of the posterior.

Result: The rank normalized effective sample size is more stable than classic effective sample size, which is not well defined for the Cauchy distribution.

Result: It is useful to look at both bulk- and tail-ESS.

We check also lp__. Although increasing max_treedepth improved bulk-ESS of x, the efficiency for lp__ didn’t change.

res <- monitor_extra(samp[, , 51:52])
cat('lp__: Bulk-ESS =', round(res['lp__', 'zsseff'], 2), '\n')

lp__: Bulk-ESS = 240

cat('lp__: Tail-ESS =', round(res['lp__', 'tailseff'], 2), '\n')

lp__: Tail-ESS = 587

We examine the sampling efficiency in different parts of the posterior by computing the effective sample size for small interval probability estimates.

plot_local_ess(fit = fit_nom_td20, par = which_min_ess, nalpha = 20)

It seems that increasing max_treedepth has not much improved the efficiency in the tails. We also examine the effective sample size of different quantile estimates.

plot_quantile_ess(fit = fit_nom_td20, par = which_min_ess, nalpha = 40)

The rank plot visualisation of x[11], which has the smallest tail-ESS of NaN among the \(x\), indicates clear convergence problems.

xmin <- paste0("x[", which_min_ess, "]")
mcmc_hist_r_scale(samp[, , xmin])

The rank plot visualisation of lp__, which has an effective sample size 240, doesn’t look so good either.

mcmc_hist_r_scale(samp[, , "lp__"])

Default Stan options + increased maximum treedepth + longer chains

Let’s try running 8 times longer chains.

fit_nom_td20l <- stan(
  file = 'cauchy_nom.stan', seed = 7878, 
  refresh = 0, control = list(max_treedepth = 20), 
  warmup = 1000, iter = 9000
)

Warning: There were 7 transitions after warmup that exceeded the maximum treedepth. Increase max_treedepth above 20. See
http://mc-stan.org/misc/warnings.html#maximum-treedepth-exceeded

Warning: Examine the pairs() plot to diagnose sampling problems

Trace plots for the first parameter still look wild with occasional large values.

samp <- as.array(fit_nom_td20l)
mcmc_trace(samp[, , 1])

res <- monitor_extra(samp[, , 1:50])
which_min_ess <- which.min(res$tailseff)

Let’s check the diagnostics for all \(x\).

plot_rhat(res)

All Rhats are below \(1.01\). The classic split-Rhat has more variation than the rank normalized Rhat (note that the former is not well defined in this case).

plot_ess(res)

Most classic ESS’s are close to zero. Running longer chains just made most classic ESS’s even smaller.

The smallest bulk-ESS are around 5000, which is not that bad. The smallest median-ESS’s are larger than 25000, that is we are able to estimate the median efficiently. However, the smallest tail-ESS is 919 indicating problems for estimating the scale of the posterior.

Result: The rank normalized effective sample size is more stable than classic effective sample size even for very long chains.

Result: It is useful to look at both bulk- and tail-ESS.

We also check lp__. Although increasing the number of iterations improved bulk-ESS of the \(x\), the relative efficiency for lp__ didn’t change.

res <- monitor_extra(samp[, , 51:52])
cat('lp__: Bulk-ESS =', round(res['lp__', 'zsseff'], 2), '\n')

lp__: Bulk-ESS = 1289

cat('lp__: Tail-ESS =', round(res['lp__', 'tailseff'], 2), '\n')

lp__: Tail-ESS = 1887

We examine the sampling efficiency in different parts of the posterior by computing the effective sample size for small interval probability estimates.

plot_local_ess(fit = fit_nom_td20l, par = which_min_ess, nalpha = 20)

Increasing the chain length did not seem to change the relative efficiency. With more draws from the longer chains we can use a finer resolution for the local efficiency estimates.

plot_local_ess(fit = fit_nom_td20l, par = which_min_ess, nalpha = 100)

The sampling efficiency far in the tails is worryingly low. This was more difficult to see previously with less draws from the tails. We see similar problems in the plot of effective sample size for quantiles.

plot_quantile_ess(fit = fit_nom_td20l, par = which_min_ess, nalpha = 100)

Let’s look at the rank plot visualisation of x[39], which has the smallest tail-ESS NaN among the \(x\).

xmin <- paste0("x[", which_min_ess, "]")
mcmc_hist_r_scale(samp[, , xmin])

Increasing the number of iterations couldn’t remove the mixing problems at the tails. The mixing problem is inherent to the nominal parameterization of Cauchy distribution.

First alternative parameterization of the Cauchy distribution

Next, we examine an alternative parameterization and consider the Cauchy distribution as a scale mixture of Gaussian distributions. The model has two parameters and the Cauchy distributed \(x\) can be computed from those. In addition to improved sampling performance, the example illustrates that focusing on diagnostics matters.

writeLines(readLines("cauchy_alt_1.stan"))

parameters {
  vector[50] x_a;
  vector<lower=0>[50] x_b;
}

transformed parameters {
  vector[50] x = x_a ./ sqrt(x_b);
}

model {
  x_a ~ normal(0, 1);
  x_b ~ gamma(0.5, 0.5);
}

generated quantities {
  real I = fabs(x[1]) < 1 ? 1 : 0;
}

We run the alternative model:

fit_alt1 <- stan(file='cauchy_alt_1.stan', seed=7878, refresh = 0)

There are no warnings and the sampling is much faster.

samp <- as.array(fit_alt1)
res <- monitor_extra(samp[, , 101:150])
which_min_ess <- which.min(res$tailseff)

plot_rhat(res)

All Rhats are below \(1.01\). Classic split-Rhats also look good even though they are not well defined for the Cauchy distribution.

plot_ess(res)

Result: Rank normalized ESS’s have less variation than classic one which is not well defined for Cauchy.

We check lp__:

res <- monitor_extra(samp[, , 151:152])
cat('lp__: Bulk-ESS =', round(res['lp__', 'zsseff'], 2), '\n')

lp__: Bulk-ESS = 1310

cat('lp__: Tail-ESS =', round(res['lp__', 'tailseff'], 2), '\n')

lp__: Tail-ESS = 1928

The relative efficiencies for lp__ are also much better than with the nominal parameterization.

We examine the sampling efficiency in different parts of the posterior by computing the effective sample size for small interval probability estimates.

plot_local_ess(fit = fit_alt1, par = 100 + which_min_ess, nalpha = 20)

The effective sample size is good in all parts of the posterior. We also examine the effective sample size of different quantile estimates.

plot_quantile_ess(fit = fit_alt1, par = 100 + which_min_ess, nalpha = 40)

We compare the mean relative efficiencies of the underlying parameters in the new parameterization and the actual \(x\) we are interested in.

res <- monitor_extra(samp[, , 101:150])
res1 <- monitor_extra(samp[, , 1:50])
res2 <- monitor_extra(samp[, , 51:100])

cat('Mean Bulk-ESS for x =' , round(mean(res[, 'zsseff']), 2), '\n')

Mean Bulk-ESS for x = 3629.24

cat('Mean Tail-ESS for x =' , round(mean(res[, 'tailseff']), 2), '\n')

Mean Tail-ESS for x = 2265.22

cat('Mean Bulk-ESS for x_a =' , round(mean(res1[, 'zsseff']), 2), '\n')

Mean Bulk-ESS for x_a = 3956.06

cat('Mean Bulk-ESS for x_b =' , round(mean(res2[, 'zsseff']), 2), '\n')

Mean Bulk-ESS for x_b = 2761.22

Result: We see that the effective sample size of the interesting \(x\) can be different from the effective sample size of the parameters \(x_a\) and \(x_b\) that we used to compute it.

The rank plot visualisation of x[40], which has the smallest tail-ESS of 1823 among the \(x\) looks better than for the nominal parameterization.

xmin <- paste0("x[", which_min_ess, "]")
mcmc_hist_r_scale(samp[, , xmin])

Similarly, the rank plot visualisation of lp__, which has a relative efficiency of -81.34, 0.23, 8.08, -95.19, -80.99, -68.66, 1288, 0.32, 1303, 1296, 1310, 0.33, 1, 1, 1, 1, 1, 2366, 0.59, 1928, 0.48, 1708, 0.43, 2912, 0.73 looks better than for the nominal parameterization.

mcmc_hist_r_scale(samp[, , "lp__"])

Another alternative parameterization of the Cauchy distribution

Another alternative parameterization is obtained by a univariate transformation as shown in the following code (see also the 3rd alternative in Michael Betancourt’s case study).

writeLines(readLines("cauchy_alt_3.stan"))

parameters {
  vector<lower=0, upper=1>[50] x_tilde;
}

transformed parameters {
vector[50] x = tan(pi() * (x_tilde - 0.5));
}

model {
  // Implicit uniform prior on x_tilde
}

generated quantities {
  real I = fabs(x[1]) < 1 ? 1 : 0;
}

We run the alternative model:

fit_alt3 <- stan(file='cauchy_alt_3.stan', seed=7878, refresh = 0)

There are no warnings, and the sampling is much faster than for the nominal model.

samp <- as.array(fit_alt3)
res <- monitor_extra(samp[, , 51:100])
which_min_ess <- which.min(res$tailseff)

plot_rhat(res)

All Rhats except some folded Rhats are below 1.01. Classic split-Rhat’s look also good even though it is not well defined for the Cauchy distribution.

plot_ess(res)

Result: Rank normalized relative efficiencies have less variation than classic ones. Bulk-ESS and median-ESS are slightly larger than 1, which is possible for antithetic Markov chains which have negative correlation for odd lags.

We also take a closer look at the lp__ value:

res <- monitor_extra(samp[, , 101:102])
cat('lp__: Bulk-ESS =', round(res['lp__', 'zsseff'], 2), '\n')

lp__: Bulk-ESS = 1494

cat('lp__: Tail-ESS =', round(res['lp__', 'tailseff'], 2), '\n')

lp__: Tail-ESS = 1884

The effective sample size for these are also much better than with the nominal parameterization.

We examine the sampling efficiency in different parts of the posterior by computing the effective sample size for small interval probability estimates.

plot_local_ess(fit = fit_alt3, par = 50 + which_min_ess, nalpha = 20)

We examine also the sampling efficiency of different quantile estimates.

plot_quantile_ess(fit = fit_alt3, par = 50 + which_min_ess, nalpha = 40)

The effective sample size in tails is worse than for the first alternative parameterization, although it’s still better than for the nominal parameterization.

We compare the mean effective sample size of the underlying parameter in the new parameterization and the actually Cauchy distributed \(x\) we are interested in.

res <- monitor_extra(samp[, , 51:100])
res1 <- monitor_extra(samp[, , 1:50])
cat('Mean bulk-seff for x =' , round(mean(res[, 'zsseff']), 2), '\n')

Mean bulk-seff for x = 4702.98

cat('Mean tail-seff for x =' , round(mean(res[, 'zfsseff']), 2), '\n')

Mean tail-seff for x = 1602.7

cat('Mean bulk-seff for x_tilde =' , round(mean(res1[, 'zsseff']), 2), '\n')

Mean bulk-seff for x_tilde = 4702.98

cat('Mean tail-seff for x_tilde =' , round(mean(res1[, 'zfsseff']), 2), '\n')

Mean tail-seff for x_tilde = 1612.14

The Rank plot visualisation of x[5], which has the smallest tail-ESS of 1891 among the \(x\) reveals shows good efficiency, similar to the results for lp__.

xmin <- paste0("x[", which_min_ess, "]")
mcmc_hist_r_scale(samp[, , xmin])

mcmc_hist_r_scale(samp[, , "lp__"])

Half-Cauchy distribution with nominal parameterization

writeLines(readLines("half_cauchy_nom.stan"))

parameters {
  vector<lower=0>[50] x;
}

model {
  x ~ cauchy(0, 1);
}

generated quantities {
  real I = fabs(x[1]) < 1 ? 1 : 0;
}

We run the half-Cauchy model with nominal parameterization (and positive constraint).

fit_half_nom <- stan(file = 'half_cauchy_nom.stan', seed = 7878, refresh = 0)

There are no warnings and the sampling is much faster than for the full Cauchy distribution with nominal parameterization.

samp <- as.array(fit_half_nom)
res <- monitor_extra(samp[, , 1:50])
which_min_ess <- which.min(res$tailseff)

plot_rhat(res)

All Rhats are below \(1.01\). Classic split-Rhats also look good even though they are not well defined for the half-Cauchy distribution.

plot_ess(res)

Result: Rank normalized effective sample size have less variation than classic ones. Some Bulk-ESS and median-ESS are larger than 1, which is possible for antithetic Markov chains which have negative correlation for odd lags.

Due to the <lower=0> constraint, the sampling was made in the log(x) space, and we can also check the performance in that space.

res <- monitor_extra(log(samp[, , 1:50]))

plot_ess(res)

\(\log(x)\) is quite close to Gaussian, and thus classic effective sample size is also close to rank normalized ESS which is exactly the same as for the original \(x\) as rank normalization is invariant to bijective transformations.

Result: The rank normalized effective sample size is close to the classic effective sample size for transformations which make the distribution close to Gaussian.

We examine the sampling efficiency in different parts of the posterior by computing the effective sample size for small interval probability estimates.

plot_local_ess(fit = fit_half_nom, par = which_min_ess, nalpha = 20)

The effective sample size is good overall, with only a small dip in tails. We can also examine the effective sample size of different quantile estimates.

plot_quantile_ess(fit = fit_half_nom, par = which_min_ess, nalpha = 40)

The rank plot visualisation of x[32], which has the smallest tail-ESS of 1742 among \(x\), looks good.

xmin <- paste0("x[", which_min_ess, "]")
mcmc_hist_r_scale(samp[, , xmin])

The rank plot visualisation of lp__ reveals some small differences in the scales, but it’s difficult to know whether this small variation from uniform is relevant.

mcmc_hist_r_scale(samp[, , "lp__"])

Alternative parameterization of the half-Cauchy distribution

writeLines(readLines("half_cauchy_alt.stan"))

parameters {
  vector<lower=0>[50] x_a;
  vector<lower=0>[50] x_b;
}

transformed parameters {
  vector[50] x = x_a .* sqrt(x_b);
}

model {
  x_a ~ normal(0, 1);
  x_b ~ inv_gamma(0.5, 0.5);
}

generated quantities {
  real I = fabs(x[1]) < 1 ? 1 : 0;
}

Run half-Cauchy with alternative parameterization

fit_half_reparam <- stan(
  file = 'half_cauchy_alt.stan', seed = 7878, refresh = 0
)

There are no warnings and the sampling is as fast for the half-Cauchy nominal model.

samp <- as.array(fit_half_reparam)
res <- monitor_extra(samp[, , 101:150])
which_min_ess <- which.min(res$tailseff)

plot_rhat(res)

plot_ess(res)

Result: The Rank normalized relative efficiencies have less variation than classic ones which is not well defined for the Cauchy distribution. Based on bulk-ESS and median-ESS, the efficiency for central quantities is much lower, but based on tail-ESS and MAD-ESS, the efficiency in the tails is slightly better than for the half-Cauchy distribution with nominal parameterization. We also see that a parameterization which is good for the full Cauchy distribution is not necessarily good for the half-Cauchy distribution as the <lower=0> constraint additionally changes the parameterization.

We also check the lp__ values:

res <- monitor_extra(samp[, , 151:152])
cat('lp__: Bulk-ESS =', round(res['lp__', 'zsseff'], 2), '\n')

lp__: Bulk-ESS = 977

cat('lp__: Tail-ESS =', round(res['lp__', 'tailseff'], 2), '\n')

lp__: Tail-ESS = 1750

We examine the sampling efficiency in different parts of the posterior by computing the effective sample size for small interval probability estimates.

plot_local_ess(fit_half_reparam, par = 100 + which_min_ess, nalpha = 20)

We also examine the effective sample size for different quantile estimates.

plot_quantile_ess(fit_half_reparam, par = 100 + which_min_ess, nalpha = 40)

The effective sample size near zero is much worse than for the half-Cauchy distribution with nominal parameterization.

The Rank plot visualisation of x[20], which has the smallest tail-ESS of NaN among the \(x\), reveals deviations from uniformity, which is expected with lower effective sample size.

xmin <- paste0("x[", which_min_ess, "]")
mcmc_hist_r_scale(samp[, , xmin])

A similar result is obtained when looking at the rank plots of lp__.

mcmc_hist_r_scale(samp[, , "lp__"])

The Cauchy distribution with Jags

So far, we have run all models in Stan, but we want to also investigate whether similar problems arise with probabilistic programming languages that use other samplers than variants of Hamiltonian Monte-Carlo. Thus, we will fit the eight schools models also with Jags, which uses a dialect of the BUGS language to specify models. Jags uses a clever mix of Gibbs and Metropolis-Hastings sampling. This kind of sampling does not scale well to high dimensional posteriors of strongly interdependent parameters, but for the relatively simple models discussed in this case study it should work just fine.

The Jags code for the nominal parameteriztion of the cauchy distribution looks as follows:

writeLines(readLines("cauchy_nom.bugs"))

model {
  for (i in 1:50) {
    x[i] ~ dt(0, 1, 1)
  }
}

First, we initialize the Jags model for reusage later.

jags_nom <- jags.model(
  "cauchy_nom.bugs",
  n.chains = 4, n.adapt = 10000
)

Compiling model graph
   Resolving undeclared variables
   Allocating nodes
Graph information:
   Observed stochastic nodes: 0
   Unobserved stochastic nodes: 50
   Total graph size: 52

Initializing model

Next, we sample 1000 iterations for each of the 4 chains for easy comparison with the corresponding Stan results.

samp_jags_nom <- coda.samples(
  jags_nom, variable.names = "x",
  n.iter = 1000
)
samp_jags_nom <- aperm(abind(samp_jags_nom, along = 3), c(1, 3, 2))
dimnames(samp_jags_nom)[[2]] <- paste0("chain:", 1:4)

We summarize the model as follows:

mon <- monitor(samp_jags_nom)
print(mon)

Inference for the input samples (4 chains: each with iter = 1000; warmup = 0):

        Q5      Q50 Q95     Mean    SD  Rhat Bulk_ESS Tail_ESS
x[1]  -6.2  0.02500 6.4   -0.770    61 1.000     4024     3825
x[2]  -7.1  0.02500 5.6    0.990    72 1.000     3697     3682
x[3]  -6.2 -0.00360 5.8   -0.470    32 1.000     4284     3896
x[4]  -6.6 -0.01100 5.6    0.210   180 1.000     3872     3734
x[5]  -5.9  0.01000 6.6    0.290    31 1.000     4282     3947
x[6]  -6.8 -0.00170 6.0    1.300    58 1.000     3994     3890
x[7]  -7.6 -0.04800 5.8   -0.190    68 1.000     3899     3889
x[8]  -6.0 -0.01200 6.5   -3.900   200 1.000     3500     3607
x[9]  -6.6 -0.00940 6.8    0.870    76 1.000     3949     3928
x[10] -6.3  0.00760 5.9   -0.520    46 1.000     4238     3932
x[11] -5.9  0.01500 6.0    0.330   120 1.000     4087     3728
x[12] -6.6  0.00420 6.0    2.600   180 1.000     3842     3787
x[13] -7.5  0.01600 6.6   -0.400    18 1.000     3938     4051
x[14] -7.1  0.02300 6.1   -5.900   170 1.000     3759     3770
x[15] -7.6 -0.04100 6.2    3.600   320 1.000     3696     3519
x[16] -6.0  0.00710 6.4    2.100    89 1.000     3570     4000
x[17] -5.7  0.05400 7.1   15.000   760 1.000     3860     3996
x[18] -6.7 -0.01800 6.1   -0.082    25 1.000     3985     3644
x[19] -6.1 -0.01800 6.5    0.910    47 1.000     3944     3972
x[20] -6.4  0.02800 6.9    0.490    28 1.000     3921     3915
x[21] -6.0  0.02600 6.3   -1.800   110 1.000     4061     4100
x[22] -6.9  0.01200 7.2    1.100    99 1.000     3835     3866
x[23] -6.4  0.02100 6.2   -0.410    19 1.000     4039     4096
x[24] -6.9 -0.06300 6.1   -0.880    33 1.000     3964     3758
x[25] -6.0  0.02500 6.3    1.800    86 1.000     4034     3930
x[26] -6.7 -0.01400 6.0 -630.000 40000 1.000     4086     3974
x[27] -7.2 -0.03800 6.0    0.029    31 1.000     3550     3763
x[28] -6.7 -0.02400 6.1    0.950    56 1.000     3631     3793
x[29] -6.2 -0.00310 5.8   -0.022    21 1.000     3996     3795
x[30] -6.1  0.01900 6.7    0.011    24 1.000     3823     3972
x[31] -6.6  0.02900 6.2  -12.000   680 1.000     3576     3664
x[32] -7.1  0.01400 7.0   -3.400   190 1.000     4260     3717
x[33] -6.1  0.03300 5.7    0.560    52 1.000     4028     3851
x[34] -6.1  0.00055 6.4    0.350    23 1.000     4010     3751
x[35] -5.9 -0.00160 6.9    3.900   200 1.000     3802     3703
x[36] -6.1 -0.03000 6.1   -0.940   280 1.000     4092     3967
x[37] -6.8 -0.00890 6.7    0.820    63 0.999     3918     3831
x[38] -5.6 -0.00630 6.1    0.032    25 1.000     3977     3751
x[39] -5.9 -0.03200 5.7    1.400    75 1.000     3795     3872
x[40] -5.9  0.02400 6.3    1.100    69 1.000     3975     3888
x[41] -5.9  0.00670 5.8   -0.360    72 1.000     4033     4147
x[42] -6.7 -0.00460 7.1   -0.024    27 1.000     4116     4143
x[43] -5.9 -0.02800 7.2   -2.400   180 1.000     3995     3846
x[44] -6.4 -0.04500 6.0    0.930    57 1.000     3978     3974
x[45] -7.0  0.00580 6.8    2.200   110 1.000     4056     3971
x[46] -6.2  0.03000 6.5    0.430    49 1.000     4020     4013
x[47] -6.3  0.00360 5.6   -0.950    38 1.000     3936     4143
x[48] -6.3 -0.01700 6.7   -0.098    24 1.000     3966     3609
x[49] -6.9 -0.00530 6.8   -0.780    80 1.000     3833     3655
x[50] -6.7 -0.00360 7.0    0.038    63 1.000     4096     3799

For each parameter, Bulk_ESS and Tail_ESS are crude measures of 
effective sample size for bulk and tail quantities respectively (good values is 
ESS > 400), and Rhat is the potential scale reduction factor on rank normalized
split chains (at convergence, Rhat = 1).

which_min_ess <- which.min(mon[1:50, 'Bulk_ESS'])

The overall results look very promising with Rhats = 1 and ESS values close to the total number of draws of 4000. We take a detailed look at x[8], which has the smallest bulk-ESS of 3500.

We examine the sampling efficiency in different parts of the posterior by computing the efficiency estimates for small interval probability estimates.

plot_local_ess(fit = samp_jags_nom, par = which_min_ess, nalpha = 20)

The efficiency estimate is good in all parts of the posterior. Further, we examine the sampling efficiency of different quantile estimates.

plot_quantile_ess(fit = samp_jags_nom, par = which_min_ess, nalpha = 40)

Rank plots also look rather similar across chains.

xmin <- paste0("x[", which_min_ess, "]")
mcmc_hist_r_scale(samp_jags_nom[, , xmin])

Result: Jags seems to be able to sample from the nominal parameterization of the Cauchy distribution just fine.

Appendix F: Hierarchical model: Eight Schools

We continue with our discussion about hierarchical models on the Eight Schools data, which we started in Section Eight Schools. We also analyse the performance of different variants of the diagnostics.

A Centered Eight Schools model

writeLines(readLines("eight_schools_cp.stan"))

data {
  int<lower=0> J;
  real y[J];
  real<lower=0> sigma[J];
}

parameters {
  real mu;
  real<lower=0> tau;
  real theta[J];
}

model {
  mu ~ normal(0, 5);
  tau ~ cauchy(0, 5);
  theta ~ normal(mu, tau);
  y ~ normal(theta, sigma);
}

In the main text, we observed that the centered parameterization of this hierarchical model did not work well with the default MCMC options of Stan plus increased adapt_delta, and so we directly try to fit the model with longer chains.

Centered parameterization with longer chains

Low efficiency can be sometimes compensated with longer chains. Let’s check 10 times longer chain.

fit_cp2 <- stan(
  file = 'eight_schools_cp.stan', data = eight_schools,
  iter = 20000, chains = 4, seed = 483892929, refresh = 0,
  control = list(adapt_delta = 0.95)
)

Warning: There were 2335 divergent transitions after warmup. Increasing adapt_delta above 0.95 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup

Warning: There were 1 chains where the estimated Bayesian Fraction of Missing Information was low. See
http://mc-stan.org/misc/warnings.html#bfmi-low

Warning: Examine the pairs() plot to diagnose sampling problems

monitor(fit_cp2)

Inference for the input samples (4 chains: each with iter = 20000; warmup = 10000):

             Q5   Q50   Q95  Mean  SD  Rhat Bulk_ESS Tail_ESS
mu        -0.99   4.8 10.00   4.9 3.6  1.05       71      189
tau        0.33   2.8 10.00   3.7 3.3  1.08       45       17
theta[1]  -1.40   6.4 16.00   6.8 5.6  1.01      407     9491
theta[2]  -2.50   5.5 13.00   5.4 4.8  1.02      153     9429
theta[3]  -4.90   4.7 12.00   4.4 5.4  1.03      117    10374
theta[4]  -2.90   5.3 12.00   5.2 5.0  1.03      140     9670
theta[5]  -4.50   4.3 11.00   4.1 4.9  1.04       89     4758
theta[6]  -4.10   4.7 11.00   4.5 5.1  1.03      118    11277
theta[7]  -0.88   6.6 16.00   6.8 5.1  1.01      449    11102
theta[8]  -3.50   5.4 13.00   5.3 5.5  1.02      172    10408
lp__     -25.00 -15.0  0.22 -14.0 7.6  1.07       50       86

For each parameter, Bulk_ESS and Tail_ESS are crude measures of 
effective sample size for bulk and tail quantities respectively (good values is 
ESS > 400), and Rhat is the potential scale reduction factor on rank normalized
split chains (at convergence, Rhat = 1).

res <- monitor_extra(fit_cp2)
print(res)

Inference for the input samples (4 chains: each with iter = 20000; warmup = 10000):

          mean se_mean  sd     Q5   Q50   Q95 seff    reff sseff zseff zsseff zsreff  Rhat sRhat
mu         4.9    0.49 3.6  -0.99   4.8 10.00   53 0.00130    71    54     71 0.0018  1.05   1.0
tau        3.7    0.30 3.3   0.33   2.8 10.00  120 0.00310   170    35     45 0.0011  1.02   1.0
theta[1]   6.8    0.22 5.6  -1.40   6.4 16.00  670 0.01700  1100   280    410 0.0100  1.01   1.0
theta[2]   5.4    0.43 4.8  -2.50   5.5 13.00  120 0.00310   170   110    150 0.0038  1.02   1.0
theta[3]   4.4    0.53 5.4  -4.90   4.7 12.00  100 0.00260   150    86    120 0.0029  1.03   1.0
theta[4]   5.2    0.46 5.0  -2.90   5.3 12.00  120 0.00300   160   100    140 0.0035  1.02   1.0
theta[5]   4.1    0.57 4.9  -4.50   4.3 11.00   76 0.00190   100    69     89 0.0022  1.03   1.0
theta[6]   4.5    0.51 5.1  -4.10   4.7 11.00  100 0.00250   140    87    120 0.0030  1.03   1.0
theta[7]   6.8    0.23 5.1  -0.88   6.6 16.00  510 0.01300   740   310    450 0.0110  1.01   1.0
theta[8]   5.3    0.43 5.5  -3.50   5.4 13.00  160 0.00400   230   120    170 0.0043  1.02   1.0
lp__     -14.0    1.30 7.6 -25.00 -15.0  0.22   33 0.00082    44    37     50 0.0012  1.08   1.1
         zRhat zsRhat zfsRhat zfsseff zfsreff tailseff tailreff medsseff medsreff madsseff madsreff
mu         1.0    1.0     1.0     150  0.0038      190  0.00470      170   0.0044      170   0.0044
tau        1.1    1.1     1.0    1000  0.0260       17  0.00042      180   0.0044      170   0.0042
theta[1]   1.0    1.0     1.0    3300  0.0820     9500  0.24000      180   0.0044      270   0.0067
theta[2]   1.0    1.0     1.0    1800  0.0450     9400  0.24000      170   0.0043      180   0.0044
theta[3]   1.0    1.0     1.0     740  0.0180    10000  0.26000      170   0.0042      170   0.0043
theta[4]   1.0    1.0     1.0    1500  0.0370     9700  0.24000      170   0.0042      170   0.0042
theta[5]   1.0    1.0     1.0     380  0.0094     4800  0.12000      170   0.0042      180   0.0044
theta[6]   1.0    1.0     1.0     640  0.0160    11000  0.28000      180   0.0044      180   0.0044
theta[7]   1.0    1.0     1.0    2800  0.0690    11000  0.28000      180   0.0045      850   0.0210
theta[8]   1.0    1.0     1.0    2800  0.0700    10000  0.26000      170   0.0042      190   0.0048
lp__       1.1    1.1     1.1      55  0.0014       86  0.00220      170   0.0042      160   0.0039

We still get a whole bunch of divergent transitions so it’s clear that the results can’t be trusted even if all other diagnostics were good. Still, it may be worth looking at additional diagnostics to better understand what’s happening.

Some rank-normalized split-Rhats are still larger than \(1.01\). Bulk-ESS for tau and lp__ are around 800 which corresponds to low relative efficiency of \(1\%\), but is above our recommendation of ESS>400. In this kind of cases, it is useful to look at the local efficiency estimates, too (and the larger number of divergences is clear indication of problems, too).

We examine the sampling efficiency in different parts of the posterior by computing the effective sample size for small intervals for tau.

plot_local_ess(fit = fit_cp2, par = "tau", nalpha = 50)

We see that the sampling has difficulties in exploring small tau values. As ESS<400 for small probability intervals in case of small tau values, we may suspect that we may miss substantial amount of posterior mass and get biased estimates.

We also examine the effective sample size of different quantile estimates.

plot_quantile_ess(fit = fit_cp2, par = "tau", nalpha = 100)

Several quantile estimates have ESS<400, which raises a doubt that there are convergence problems and we may have significant bias.

Let’s see how the Bulk-ESS and Tail-ESS changes when we use more and more draws.

plot_change_ess(fit = fit_cp2, par = "tau")

We see that given recommendation that Bulk-ESS>400 and Tail-ESS>400, they are not sufficient to detect convergence problems in this case, even the tail quantile estimates are able to detect these problems.

The rank plot visualisation of tau also shows clear sticking and mixing problems.

samp_cp2 <- as.array(fit_cp2)
mcmc_hist_r_scale(samp_cp2[, , "tau"])

Similar results are obtained for lp__, which is closely connected to tau for this model.

mcmc_hist_r_scale(samp_cp2[, , "lp__"])

We may also examine small interval efficiencies for mu.

plot_local_ess(fit = fit_cp2, par = "mu", nalpha = 50)

There are gaps of poor efficiency which again indicates problems in the mixing of the chains. However, these problems do not occur for any specific range of values of mu as was the case for tau. This tells us that it’s probably not mu with which the sampler has problems, but more likely tau or a related quantity.

As we observed divergences, we shouldn’t trust any Monte Carlo standard error (MCSE) estimates as they are likely biased, as well. However, for illustration purposes, we compute the MCSE, tail quantiles and corresponding effective sample sizes for the median of mu and tau. Comparing to the shorter MCMC run, using 10 times more draws has not reduced the MCSE to one third as would be expected without problems in the mixing of the chains.

round(quantile_mcse(samp_cp2[ , , "mu"], prob = 0.5), 2)

  mcse  Q05  Q95   Seff
1 0.37 4.22 5.43 173.52

round(quantile_mcse(samp_cp2[ , , "tau"], prob = 0.5), 2)

  mcse  Q05  Q95   Seff
1 0.27 2.38 3.27 174.86

Centered parameterization with very long chains

For further evidence, let’s check 100 times longer chains than the default. This is not something we would recommend doing in practice, as it is not able to solve any problems with divergences as illustrated below.

fit_cp3 <- stan(
  file = 'eight_schools_cp.stan', data = eight_schools,
  iter = 200000, chains = 4, seed = 483892929, refresh = 0,
  control = list(adapt_delta = 0.95)
)

Warning: There were 11699 divergent transitions after warmup. Increasing adapt_delta above 0.95 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup

Warning: There were 3 chains where the estimated Bayesian Fraction of Missing Information was low. See
http://mc-stan.org/misc/warnings.html#bfmi-low

Warning: Examine the pairs() plot to diagnose sampling problems

monitor(fit_cp3)

Inference for the input samples (4 chains: each with iter = 2e+05; warmup = 1e+05):

             Q5   Q50  Q95  Mean  SD  Rhat Bulk_ESS Tail_ESS
mu        -1.10   4.4  9.8   4.4 3.3     1    18335    30265
tau        0.47   2.9 10.0   3.8 3.2     1     2200      769
theta[1]  -1.60   5.7 16.0   6.3 5.7     1    23832   110854
theta[2]  -2.50   4.9 13.0   4.9 4.8     1    27789   136002
theta[3]  -5.10   4.1 12.0   3.9 5.4     1    39355   122761
theta[4]  -3.00   4.7 13.0   4.7 4.9     1    32607   138545
theta[5]  -4.60   3.8 11.0   3.6 4.7     1    34479    44492
theta[6]  -4.20   4.2 12.0   4.0 4.9     1    37000    92227
theta[7]  -1.00   5.9 16.0   6.4 5.2     1    20685    58049
theta[8]  -3.50   4.7 14.0   4.9 5.4     1    36212   125498
lp__     -25.00 -15.0 -2.1 -15.0 6.9     1     2541     1074

For each parameter, Bulk_ESS and Tail_ESS are crude measures of 
effective sample size for bulk and tail quantities respectively (good values is 
ESS > 400), and Rhat is the potential scale reduction factor on rank normalized
split chains (at convergence, Rhat = 1).

res <- monitor_extra(fit_cp3)
print(res)

Inference for the input samples (4 chains: each with iter = 2e+05; warmup = 1e+05):

          mean se_mean  sd     Q5   Q50  Q95  seff  reff sseff zseff zsseff zsreff  Rhat sRhat
mu         4.4   0.024 3.3  -1.10   4.4  9.8 18000 0.046 18000 18000  18000 0.0460     1     1
tau        3.8   0.033 3.2   0.47   2.9 10.0  9400 0.023  9400  2200   2200 0.0055     1     1
theta[1]   6.3   0.032 5.7  -1.60   5.7 16.0 31000 0.077 31000 24000  24000 0.0600     1     1
theta[2]   4.9   0.026 4.8  -2.50   4.9 13.0 33000 0.082 33000 28000  28000 0.0690     1     1
theta[3]   3.9   0.023 5.4  -5.10   4.1 12.0 54000 0.130 54000 39000  39000 0.0980     1     1
theta[4]   4.7   0.024 4.9  -3.00   4.7 13.0 39000 0.099 40000 33000  33000 0.0820     1     1
theta[5]   3.6   0.023 4.7  -4.60   3.8 11.0 41000 0.100 42000 34000  34000 0.0860     1     1
theta[6]   4.0   0.023 4.9  -4.20   4.2 12.0 46000 0.110 46000 37000  37000 0.0920     1     1
theta[7]   6.4   0.033 5.2  -1.00   5.9 16.0 25000 0.062 25000 21000  21000 0.0520     1     1
theta[8]   4.9   0.024 5.4  -3.50   4.7 14.0 50000 0.130 50000 36000  36000 0.0910     1     1
lp__     -15.0   0.140 6.9 -25.00 -15.0 -2.1  2400 0.006  2400  2500   2500 0.0064     1     1
         zRhat zsRhat zfsRhat zfsseff zfsreff tailseff tailreff medsseff medsreff madsseff madsreff
mu           1      1       1   29000  0.0720    30000   0.0760    16000    0.041    18000    0.046
tau          1      1       1   33000  0.0820      770   0.0019    11000    0.027    15000    0.038
theta[1]     1      1       1   41000  0.1000   110000   0.2800    15000    0.038    19000    0.047
theta[2]     1      1       1   42000  0.1000   140000   0.3400    15000    0.038    18000    0.046
theta[3]     1      1       1   31000  0.0780   120000   0.3100    17000    0.042    22000    0.055
theta[4]     1      1       1   39000  0.0970   140000   0.3500    17000    0.041    19000    0.048
theta[5]     1      1       1   37000  0.0930    44000   0.1100    17000    0.041    19000    0.047
theta[6]     1      1       1   37000  0.0910    92000   0.2300    16000    0.041    18000    0.045
theta[7]     1      1       1   35000  0.0880    58000   0.1500    15000    0.037    17000    0.043
theta[8]     1      1       1   39000  0.0970   130000   0.3100    15000    0.039    16000    0.040
lp__         1      1       1    2900  0.0074     1100   0.0027    10000    0.025    14000    0.034

Rhat, Bulk-ESS and Tail-ESS are not able to detect problems, although Tail-ESS for tau is suspiciously low compared to total number of draws.

plot_local_ess(fit = fit_cp3, par = "tau", nalpha = 100)

plot_quantile_ess(fit = fit_cp3, par = "tau", nalpha = 100)

And the rank plots of tau also show sticking and mixing problems for small values of tau.

samp_cp3 <- as.array(fit_cp3)
mcmc_hist_r_scale(samp_cp3[, , "tau"])

What we do see is an advantage of rank plots over trace plots as even with 100000 draws per chain, rank plots don’t get crowded and the mixing problems of chains is still easy to see.

With centered parameterization the mean estimate tends to get smaller with more draws. With 400000 draws using the centered parameterization the mean estimate is 3.77 (se 0.03). With 40000 draws using the non-centered parameterization the mean estimate is 3.6 (se 0.02). The difference is more than 8 sigmas. We are able to see the convergence problems in the centered parameterization case, if we do look carefully (or use divergence diagnostic ), but we do see that Rhat, Bulk-ESS, Tail-ESS and Monte Carlo error estimates for the mean can’t be trusted if other diagnostics indicate convergence problems!

Centered parameterization with very long chains and thinning

When autocorrelation time is high, it has been common to thin the chains by saving only a small portion of the draws. This will throw away useful information also for convergence diagnostics. With 400000 iterations per chain, thinning of 200 and 4 chains, we again end up with 4000 iterations as with the default settings.

fit_cp4 <- stan(
  file = 'eight_schools_cp.stan', data = eight_schools,
  iter = 400000, thin = 200, chains = 4, seed = 483892929, refresh = 0,
  control = list(adapt_delta = 0.95)
)

Warning: There were 93 divergent transitions after warmup. Increasing adapt_delta above 0.95 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup

Warning: There were 3 chains where the estimated Bayesian Fraction of Missing Information was low. See
http://mc-stan.org/misc/warnings.html#bfmi-low

Warning: Examine the pairs() plot to diagnose sampling problems

We observe several divergent transitions and the estimated Bayesian fraction of missing information is also low, which indicate convergence problems and potentially biased estimates.

Unfortunately the thinning makes Rhat and ESS estimates to miss the problems. The posterior mean is still biased, being more than 3 sigmas away from the estimate obtained using non-centered parameterization.

monitor(fit_cp4)

Inference for the input samples (4 chains: each with iter = 4e+05; warmup = 2e+05):

             Q5   Q50  Q95  Mean  SD  Rhat Bulk_ESS Tail_ESS
mu        -0.91   4.5  9.7   4.4 3.2     1     3784     3648
tau        0.46   2.9 10.0   3.7 3.2     1     3625     2447
theta[1]  -1.70   5.6 16.0   6.2 5.7     1     4101     3691
theta[2]  -2.20   4.8 13.0   5.0 4.6     1     3950     3946
theta[3]  -4.50   4.2 12.0   4.0 5.2     1     4121     3819
theta[4]  -3.00   4.7 12.0   4.7 4.8     1     4026     4188
theta[5]  -4.40   3.8 11.0   3.6 4.7     1     3790     3839
theta[6]  -3.80   4.3 12.0   4.2 4.9     1     4057     4059
theta[7]  -0.96   5.9 15.0   6.3 5.0     1     4154     3813
theta[8]  -3.50   4.6 13.0   4.8 5.3     1     4040     3968
lp__     -25.00 -15.0 -1.6 -14.0 7.0     1     3689     2616

For each parameter, Bulk_ESS and Tail_ESS are crude measures of 
effective sample size for bulk and tail quantities respectively (good values is 
ESS > 400), and Rhat is the potential scale reduction factor on rank normalized
split chains (at convergence, Rhat = 1).

res <- monitor_extra(fit_cp4)
print(res)

Inference for the input samples (4 chains: each with iter = 4e+05; warmup = 2e+05):

          mean se_mean  sd     Q5   Q50  Q95 seff reff sseff zseff zsseff zsreff  Rhat sRhat zRhat
mu         4.4   0.053 3.2  -0.91   4.5  9.7 3700 0.93  3800  3700   3800   0.95     1     1     1
tau        3.7   0.050 3.2   0.46   2.9 10.0 4000 1.00  4000  3600   3600   0.91     1     1     1
theta[1]   6.2   0.090 5.7  -1.70   5.6 16.0 4100 1.00  4100  4100   4100   1.00     1     1     1
theta[2]   5.0   0.074 4.6  -2.20   4.8 13.0 3900 0.98  3900  3900   4000   0.99     1     1     1
theta[3]   4.0   0.081 5.2  -4.50   4.2 12.0 4100 1.00  4100  4100   4100   1.00     1     1     1
theta[4]   4.7   0.077 4.8  -3.00   4.7 12.0 4000 0.99  4000  4000   4000   1.00     1     1     1
theta[5]   3.6   0.077 4.7  -4.40   3.8 11.0 3700 0.93  3800  3700   3800   0.95     1     1     1
theta[6]   4.2   0.077 4.9  -3.80   4.3 12.0 3900 0.99  4000  4000   4100   1.00     1     1     1
theta[7]   6.3   0.078 5.0  -0.96   5.9 15.0 4100 1.00  4100  4100   4200   1.00     1     1     1
theta[8]   4.8   0.085 5.3  -3.50   4.6 13.0 4000 0.99  4000  4000   4000   1.00     1     1     1
lp__     -14.0   0.120 7.0 -25.00 -15.0 -1.6 3500 0.88  3600  3700   3700   0.92     1     1     1
         zsRhat zfsRhat zfsseff zfsreff tailseff tailreff medsseff medsreff madsseff madsreff
mu            1       1    3700    0.91     3600     0.91     4200     1.00     3800     0.95
tau           1       1    4100    1.00     2400     0.61     4000     0.99     3800     0.95
theta[1]      1       1    4200    1.00     3700     0.92     4100     1.00     3900     0.98
theta[2]      1       1    4000    1.00     3900     0.99     3600     0.89     4100     1.00
theta[3]      1       1    3800    0.95     3800     0.95     4100     1.00     3800     0.95
theta[4]      1       1    3900    0.98     4200     1.00     3500     0.87     3900     0.97
theta[5]      1       1    3600    0.90     3800     0.96     3800     0.96     3800     0.96
theta[6]      1       1    3900    0.97     4100     1.00     4000     1.00     3800     0.96
theta[7]      1       1    4100    1.00     3800     0.95     4300     1.10     3900     0.97
theta[8]      1       1    3800    0.94     4000     0.99     4100     1.00     3800     0.96
lp__          1       1    3200    0.80     2600     0.65     3800     0.96     3900     0.98

Various diagnostic plots of tau look reasonable as well.

plot_local_ess(fit = fit_cp4, par = "tau", nalpha = 100)

plot_quantile_ess(fit = fit_cp4, par = "tau", nalpha = 100)

plot_change_ess(fit = fit_cp4, par = "tau")

However, the rank plots seem still to show the problem.

samp_cp4 <- as.array(fit_cp4)
mcmc_hist_r_scale(samp_cp4[, , "tau"])

Non-centered Eight Schools model

In the following, we want to expand our understanding of the non-centered parameterization of the hierarchical model fit to the eight schools data.

writeLines(readLines("eight_schools_ncp.stan"))

data {
  int<lower=0> J;
  real y[J];
  real<lower=0> sigma[J];
}

parameters {
  real mu;
  real<lower=0> tau;
  real theta_tilde[J];
}

transformed parameters {
  real theta[J];
  for (j in 1:J)
    theta[j] = mu + tau * theta_tilde[j];
}

model {
  mu ~ normal(0, 5);
  tau ~ cauchy(0, 5);
  theta_tilde ~ normal(0, 1);
  y ~ normal(theta, sigma);
}

Non-centered parameterization with default MCMC options

In the main text, we have already seen that the non-centered parameterization works better than the centered parameterization, at least when we use an increased adapt_delta value. Let’s see what happens when using the default MCMC option of Stan.

fit_ncp <- stan(
  file = 'eight_schools_ncp.stan', data = eight_schools,
  iter = 2000, chains = 4, seed = 483892929, refresh = 0
)

Warning: There were 2 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup

Warning: Examine the pairs() plot to diagnose sampling problems

We observe a few divergent transitions with the default of adapt_delta=0.8. Let’s analyze the sample.

monitor(fit_ncp)

Inference for the input samples (4 chains: each with iter = 2000; warmup = 1000):

                   Q5    Q50  Q95   Mean   SD  Rhat Bulk_ESS Tail_ESS
mu              -0.98  4.400  9.5  4.400 3.20     1     4083     2378
tau              0.25  2.800  9.8  3.600 3.20     1     2303     1795
theta_tilde[1]  -1.30  0.350  1.9  0.320 0.97     1     4571     2604
theta_tilde[2]  -1.40  0.140  1.6  0.120 0.92     1     5771     3078
theta_tilde[3]  -1.60 -0.100  1.5 -0.089 0.96     1     4966     3054
theta_tilde[4]  -1.40  0.033  1.5  0.046 0.91     1     5442     2830
theta_tilde[5]  -1.70 -0.170  1.4 -0.160 0.91     1     4273     3005
theta_tilde[6]  -1.60 -0.084  1.5 -0.075 0.95     1     5192     2981
theta_tilde[7]  -1.20  0.390  1.9  0.360 0.97     1     3898     2800
theta_tilde[8]  -1.50  0.073  1.7  0.079 0.97     1     4848     2863
theta[1]        -1.40  5.700 16.0  6.300 5.60     1     3790     2549
theta[2]        -2.30  4.900 13.0  5.000 4.60     1     5002     2920
theta[3]        -4.30  4.100 12.0  3.900 5.20     1     4001     3036
theta[4]        -2.70  4.700 12.0  4.600 4.60     1     4699     3063
theta[5]        -4.10  3.900 10.0  3.600 4.50     1     4310     3184
theta[6]        -4.10  4.200 11.0  4.000 4.90     1     4965     2806
theta[7]        -0.84  5.900 15.0  6.300 4.90     1     4599     3296
theta[8]        -3.20  4.800 14.0  4.900 5.40     1     4461     3288
lp__           -11.00 -6.500 -3.7 -6.800 2.30     1     1711     2385

For each parameter, Bulk_ESS and Tail_ESS are crude measures of 
effective sample size for bulk and tail quantities respectively (good values is 
ESS > 400), and Rhat is the potential scale reduction factor on rank normalized
split chains (at convergence, Rhat = 1).

res <- monitor_extra(fit_ncp)
print(res)

Inference for the input samples (4 chains: each with iter = 2000; warmup = 1000):

                 mean se_mean   sd     Q5    Q50  Q95 seff reff sseff zseff zsseff zsreff  Rhat
mu              4.400   0.051 3.20  -0.98  4.400  9.5 4000 1.00  4000  4100   4100   1.00     1
tau             3.600   0.061 3.20   0.25  2.800  9.8 2700 0.67  2700  2300   2300   0.58     1
theta_tilde[1]  0.320   0.014 0.97  -1.30  0.350  1.9 4500 1.10  4600  4500   4600   1.10     1
theta_tilde[2]  0.120   0.012 0.92  -1.40  0.140  1.6 5700 1.40  5800  5800   5800   1.40     1
theta_tilde[3] -0.089   0.014 0.96  -1.60 -0.100  1.5 4900 1.20  5000  4900   5000   1.20     1
theta_tilde[4]  0.046   0.012 0.91  -1.40  0.033  1.5 5400 1.30  5400  5400   5400   1.40     1
theta_tilde[5] -0.160   0.014 0.91  -1.70 -0.170  1.4 4200 1.10  4300  4200   4300   1.10     1
theta_tilde[6] -0.075   0.013 0.95  -1.60 -0.084  1.5 5200 1.30  5200  5200   5200   1.30     1
theta_tilde[7]  0.360   0.016 0.97  -1.20  0.390  1.9 3900 0.97  3900  3900   3900   0.97     1
theta_tilde[8]  0.079   0.014 0.97  -1.50  0.073  1.7 4800 1.20  4900  4800   4800   1.20     1
theta[1]        6.300   0.095 5.60  -1.40  5.700 16.0 3500 0.88  3500  3800   3800   0.95     1
theta[2]        5.000   0.066 4.60  -2.30  4.900 13.0 4900 1.20  4900  5000   5000   1.30     1
theta[3]        3.900   0.085 5.20  -4.30  4.100 12.0 3800 0.95  3800  4000   4000   1.00     1
theta[4]        4.600   0.069 4.60  -2.70  4.700 12.0 4600 1.10  4600  4700   4700   1.20     1
theta[5]        3.600   0.071 4.50  -4.10  3.900 10.0 4100 1.00  4200  4300   4300   1.10     1
theta[6]        4.000   0.071 4.90  -4.10  4.200 11.0 4700 1.20  4800  4900   5000   1.20     1
theta[7]        6.300   0.074 4.90  -0.84  5.900 15.0 4400 1.10  4400  4500   4600   1.10     1
theta[8]        4.900   0.084 5.40  -3.20  4.800 14.0 4000 1.00  4100  4400   4500   1.10     1
lp__           -6.800   0.056 2.30 -11.00 -6.500 -3.7 1700 0.42  1700  1700   1700   0.43     1
               sRhat zRhat zsRhat zfsRhat zfsseff zfsreff tailseff tailreff medsseff medsreff
mu                 1     1      1       1    1800    0.44     2400     0.59     4200     1.10
tau                1     1      1       1    3100    0.78     1800     0.45     3200     0.79
theta_tilde[1]     1     1      1       1    2100    0.54     2600     0.65     4500     1.10
theta_tilde[2]     1     1      1       1    2000    0.50     3100     0.77     5400     1.30
theta_tilde[3]     1     1      1       1    2100    0.53     3100     0.76     5400     1.30
theta_tilde[4]     1     1      1       1    2000    0.50     2800     0.71     4800     1.20
theta_tilde[5]     1     1      1       1    2300    0.57     3000     0.75     4300     1.10
theta_tilde[6]     1     1      1       1    2100    0.52     3000     0.75     4800     1.20
theta_tilde[7]     1     1      1       1    2300    0.56     2800     0.70     3800     0.96
theta_tilde[8]     1     1      1       1    1900    0.48     2900     0.72     4400     1.10
theta[1]           1     1      1       1    2500    0.63     2500     0.64     4400     1.10
theta[2]           1     1      1       1    2300    0.58     2900     0.73     4200     1.10
theta[3]           1     1      1       1    2600    0.64     3000     0.76     4000     1.00
theta[4]           1     1      1       1    2700    0.66     3100     0.77     4500     1.10
theta[5]           1     1      1       1    2800    0.70     3200     0.80     4500     1.10
theta[6]           1     1      1       1    2700    0.67     2800     0.70     4800     1.20
theta[7]           1     1      1       1    2500    0.62     3300     0.82     4300     1.10
theta[8]           1     1      1       1    2500    0.62     3300     0.82     4400     1.10
lp__               1     1      1       1    2500    0.62     2400     0.60     2000     0.50
               madsseff madsreff
mu                 2300     0.59
tau                3200     0.79
theta_tilde[1]     2500     0.62
theta_tilde[2]     2400     0.60
theta_tilde[3]     2400     0.61
theta_tilde[4]     2400     0.60
theta_tilde[5]     2700     0.67
theta_tilde[6]     2400     0.60
theta_tilde[7]     2800     0.70
theta_tilde[8]     2000     0.50
theta[1]           2800     0.70
theta[2]           2500     0.64
theta[3]           3000     0.74
theta[4]           2900     0.71
theta[5]           2700     0.67
theta[6]           3200     0.79
theta[7]           2900     0.73
theta[8]           2600     0.66
lp__               2800     0.70

All Rhats are close to 1, and ESSs are good despite a few divergent transitions. Small interval and quantile plots of tau reveal some sampling problems for small tau values, but not nearly as strong as for the centered parameterization.

plot_local_ess(fit = fit_ncp, par = "tau", nalpha = 20)

plot_quantile_ess(fit = fit_ncp, par = "tau", nalpha = 40)

Overall, the non-centered parameterization looks good even for the default settings of adapt_delta, and increasing it to 0.95 gets rid of the last remaining problems. This stands in sharp contrast to what we observed for the centered parameterization, where increasing adapt_delta didn’t help at all. Actually, this is something we observe quite often: A suboptimal parameterization can cause problems that are not simply solved by tuning the sampler. Instead, we have to adjust our model to achieve trustworthy inference.

Eight Schools with Jags

We will also run the centered and non-centered parameterizations of the eight schools model with Jags.

Centered Eight Schools Model

The Jags code for the centered eight schools model looks as follows:

writeLines(readLines("eight_schools_cp.bugs"))

model {
  for (j in 1:J) {
    sigma_prec[j] <- pow(sigma[j], -2)
    theta[j] ~ dnorm(mu, tau_prec)
    y[j] ~ dnorm(theta[j], sigma_prec[j])
  }
  mu ~ dnorm(0, pow(5, -2))
  tau ~ dt(0, pow(5, -2), 1)T(0, )
  tau_prec <- pow(tau, -2)
}

First, we initialize the Jags model for reusage later.

jags_cp <- jags.model(
  "eight_schools_cp.bugs",
  data = eight_schools,
  n.chains = 4, n.adapt = 10000
)

Compiling model graph
   Resolving undeclared variables
   Allocating nodes
Graph information:
   Observed stochastic nodes: 8
   Unobserved stochastic nodes: 10
   Total graph size: 40

Initializing model

Next, we sample 1000 iterations for each of the 4 chains for easy comparison with the corresponding Stan results.

samp_jags_cp <- coda.samples(
  jags_cp, c("theta", "mu", "tau"),
  n.iter = 1000
)
samp_jags_cp <- aperm(abind(samp_jags_cp, along = 3), c(1, 3, 2))

Convergence diagnostics indicate problems in the sampling of mu and tau, but also to a lesser degree in all other paramters.

mon <- monitor(samp_jags_cp)
print(mon)

Inference for the input samples (4 chains: each with iter = 1000; warmup = 0):

            Q5 Q50  Q95 Mean  SD  Rhat Bulk_ESS Tail_ESS
mu       -1.50 4.2  8.9  4.0 3.2  1.01      200      780
tau       0.28 3.5 12.0  4.5 4.4  1.03       73      100
theta[1] -1.80 5.5 19.0  6.6 6.6  1.02      272      488
theta[2] -3.00 4.8 13.0  4.8 4.9  1.01      368     1272
theta[3] -6.70 3.8 12.0  3.4 5.8  1.02      371      956
theta[4] -3.50 4.6 13.0  4.5 5.1  1.01      434     1238
theta[5] -5.90 3.5 10.0  3.0 4.9  1.02      370     1118
theta[6] -5.20 3.9 11.0  3.6 5.2  1.02      498     1220
theta[7] -1.20 5.7 17.0  6.6 5.6  1.02      247      701
theta[8] -4.30 4.8 15.0  4.9 6.1  1.01      438      809

For each parameter, Bulk_ESS and Tail_ESS are crude measures of 
effective sample size for bulk and tail quantities respectively (good values is 
ESS > 400), and Rhat is the potential scale reduction factor on rank normalized
split chains (at convergence, Rhat = 1).

We also see problems in the sampling of tau using various diagnostic plots.

plot_local_ess(samp_jags_cp, par = "tau", nalpha = 20)

plot_quantile_ess(samp_jags_cp, par = "tau", nalpha = 20)

plot_change_ess(samp_jags_cp, par = "tau")

Let’s see what happens if we run 10 times longer chains.

samp_jags_cp <- coda.samples(
  jags_cp, c("theta", "mu", "tau"),
  n.iter = 10000
)
samp_jags_cp <- aperm(abind(samp_jags_cp, along = 3), c(1, 3, 2))

Convergence looks better now, although tau is still estimated not very efficiently.

mon <- monitor(samp_jags_cp)
print(mon)

Inference for the input samples (4 chains: each with iter = 10000; warmup = 0):

            Q5 Q50  Q95 Mean  SD  Rhat Bulk_ESS Tail_ESS
mu       -1.10 4.4 10.0  4.4 3.4  1.01     1220     1620
tau       0.27 2.8  9.6  3.6 3.1  1.00      798      773
theta[1] -1.60 5.7 16.0  6.2 5.6  1.00     2041     3134
theta[2] -2.50 4.9 13.0  4.9 4.7  1.00     2219     5648
theta[3] -4.70 4.2 12.0  4.0 5.2  1.00     2234     6156
theta[4] -2.90 4.7 13.0  4.8 4.8  1.00     2230     6010
theta[5] -4.40 3.9 11.0  3.7 4.7  1.00     1847     3585
theta[6] -4.10 4.2 12.0  4.1 4.9  1.00     2150     5694
theta[7] -1.00 5.8 15.0  6.3 5.1  1.00     1895     2438
theta[8] -3.30 4.8 13.0  4.9 5.3  1.00     2439     7171

For each parameter, Bulk_ESS and Tail_ESS are crude measures of 
effective sample size for bulk and tail quantities respectively (good values is 
ESS > 400), and Rhat is the potential scale reduction factor on rank normalized
split chains (at convergence, Rhat = 1).

The diagnostic plots of quantiles and small intervals tell a similar story.

plot_local_ess(samp_jags_cp, par = "tau", nalpha = 20)

plot_quantile_ess(samp_jags_cp, par = "tau", nalpha = 20)

Notably, however, the increase in effective sample size of tau is linear in the total number of draws indicating that convergence for tau may be achieved by simply running longer chains.

plot_change_ess(samp_jags_cp, par = "tau")

Result: Similar to Stan, Jags also has convergence problems with the centered parameterization of the eight schools model.

Non-Centered Eight Schools Model

The Jags code for the non-centered eight schools model looks as follows:

writeLines(readLines("eight_schools_ncp.bugs"))

model {
  for (j in 1:J) {
    sigma_prec[j] <- pow(sigma[j], -2)
    theta_tilde[j] ~ dnorm(0, 1)
    theta[j] = mu + tau * theta_tilde[j]
    y[j] ~ dnorm(theta[j], sigma_prec[j])
  }
  mu ~ dnorm(0, pow(5, -2))
  tau ~ dt(0, pow(5, -2), 1)T(0, )
}

First, we initialize the Jags model for reusage later.

jags_ncp <- jags.model(
  "eight_schools_ncp.bugs",
  data = eight_schools,
  n.chains = 4, n.adapt = 10000
)

Compiling model graph
   Resolving undeclared variables
   Allocating nodes
Graph information:
   Observed stochastic nodes: 8
   Unobserved stochastic nodes: 10
   Total graph size: 55

Initializing model

Next, we sample 1000 iterations for each of the 4 chains for easy comparison with the corresponding Stan results.

samp_jags_ncp <- coda.samples(
  jags_ncp, c("theta", "mu", "tau"),
  n.iter = 1000
)
samp_jags_ncp <- aperm(abind(samp_jags_ncp, along = 3), c(1, 3, 2))

Convergence diagnostics indicate much better mixing than for the centered eight school model.

mon <- monitor(samp_jags_ncp)
print(mon)

Inference for the input samples (4 chains: each with iter = 1000; warmup = 0):

            Q5 Q50  Q95 Mean  SD  Rhat Bulk_ESS Tail_ESS
mu       -0.97 4.5  9.8  4.5 3.3     1     3455     3062
tau       0.24 2.6  9.1  3.4 2.9     1     1345     1831
theta[1] -1.50 5.6 16.0  6.2 5.5     1     3123     2666
theta[2] -2.50 4.8 13.0  4.9 4.6     1     4498     3779
theta[3] -4.50 4.2 12.0  4.0 5.1     1     3724     2868
theta[4] -2.40 4.9 12.0  4.8 4.6     1     4088     3279
theta[5] -4.30 4.0 10.0  3.7 4.6     1     3457     3235
theta[6] -3.60 4.3 11.0  4.1 4.7     1     4200     3849
theta[7] -0.77 5.8 15.0  6.3 4.9     1     3117     3109
theta[8] -3.10 4.8 13.0  4.9 5.2     1     4019     3267

For each parameter, Bulk_ESS and Tail_ESS are crude measures of 
effective sample size for bulk and tail quantities respectively (good values is 
ESS > 400), and Rhat is the potential scale reduction factor on rank normalized
split chains (at convergence, Rhat = 1).

Specifically, the mixing of tau looks much better although we still see some problems in the estimation of larger quantiles.

plot_local_ess(samp_jags_ncp, par = "tau", nalpha = 20)

plot_quantile_ess(samp_jags_ncp, par = "tau", nalpha = 20)

Change in effective sample size is roughly linear indicating that some remaining convergence problems are likely to be solved by running longer chains.

plot_change_ess(samp_jags_ncp, par = "tau")

Result: Similar to Stan, Jags can sample from the non-centered parameterization of the eight schools model much better than from the centered parameterization.

Appendix G: Dynamic HMC and effective sample size

We have already seen that the effective sample size of dynamic HMC can be higher than with independent draws. The next example illustrates interesting relative efficiency phenomena due to the properties of dynamic HMC algorithms.

We sample from a simple 16-dimensional standard normal model.

writeLines(readLines("normal.stan"))

data {
  int<lower=1> J;
}
parameters {
  vector[J] x;
}
model {
  x ~ normal(0, 1);
}

fit_n <- stan(
  file = 'normal.stan', data = data.frame(J = 16),
  iter = 20000, chains = 4, seed = 483892929, refresh = 0 
)

samp <- as.array(fit_n)
monitor(samp)

Inference for the input samples (4 chains: each with iter = 10000; warmup = 0):

         Q5      Q50  Q95    Mean   SD  Rhat Bulk_ESS Tail_ESS
x[1]   -1.7 -0.00240  1.6 -0.0024 1.00     1    98264    28709
x[2]   -1.6 -0.00640  1.6 -0.0031 1.00     1    95812    29664
x[3]   -1.6  0.00200  1.6  0.0031 0.99     1    98640    28669
x[4]   -1.6  0.00480  1.7  0.0055 1.00     1    97302    29166
x[5]   -1.6 -0.00091  1.6  0.0017 1.00     1   101542    29930
x[6]   -1.6 -0.00140  1.6 -0.0022 1.00     1    96292    28376
x[7]   -1.6  0.00790  1.6  0.0027 0.99     1    96016    29238
x[8]   -1.6 -0.00660  1.6 -0.0049 1.00     1   100375    29893
x[9]   -1.6  0.00590  1.7  0.0039 1.00     1   101141    28621
x[10]  -1.6 -0.00680  1.6 -0.0016 0.99     1   103126    29411
x[11]  -1.6  0.00580  1.7  0.0036 1.00     1    95886    28488
x[12]  -1.6  0.00470  1.6  0.0061 0.99     1    98433    29228
x[13]  -1.6  0.00950  1.6  0.0029 0.99     1    98181    27421
x[14]  -1.6 -0.00230  1.6 -0.0033 0.99     1    97313    27507
x[15]  -1.6  0.00520  1.6  0.0061 0.99     1    95223    29139
x[16]  -1.7  0.00310  1.6 -0.0006 1.00     1    99980    29639
lp__  -13.0 -7.70000 -3.9 -8.0000 2.80     1    14489    19627

For each parameter, Bulk_ESS and Tail_ESS are crude measures of 
effective sample size for bulk and tail quantities respectively (good values is 
ESS > 400), and Rhat is the potential scale reduction factor on rank normalized
split chains (at convergence, Rhat = 1).

res <- monitor_extra(samp)
print(res)

Inference for the input samples (4 chains: each with iter = 10000; warmup = 0):

         mean se_mean   sd    Q5      Q50  Q95   seff reff  sseff  zseff zsseff zsreff  Rhat sRhat
x[1]  -0.0024  0.0032 1.00  -1.7 -0.00240  1.6  98000 2.40  98000  98000  98000   2.50     1     1
x[2]  -0.0031  0.0032 1.00  -1.6 -0.00640  1.6  95000 2.40  96000  96000  96000   2.40     1     1
x[3]   0.0031  0.0032 0.99  -1.6  0.00200  1.6  98000 2.50  99000  98000  99000   2.50     1     1
x[4]   0.0055  0.0032 1.00  -1.6  0.00480  1.7  97000 2.40  97000  97000  97000   2.40     1     1
x[5]   0.0017  0.0031 1.00  -1.6 -0.00091  1.6 100000 2.50 100000 100000 100000   2.50     1     1
x[6]  -0.0022  0.0032 1.00  -1.6 -0.00140  1.6  96000 2.40  96000  96000  96000   2.40     1     1
x[7]   0.0027  0.0032 0.99  -1.6  0.00790  1.6  96000 2.40  96000  95000  96000   2.40     1     1
x[8]  -0.0049  0.0032 1.00  -1.6 -0.00660  1.6 100000 2.50 100000 100000 100000   2.50     1     1
x[9]   0.0039  0.0031 1.00  -1.6  0.00590  1.7 100000 2.50 100000 100000 100000   2.50     1     1
x[10] -0.0016  0.0031 0.99  -1.6 -0.00680  1.6 100000 2.60 100000 100000 100000   2.60     1     1
x[11]  0.0036  0.0032 1.00  -1.6  0.00580  1.7  95000 2.40  96000  95000  96000   2.40     1     1
x[12]  0.0061  0.0032 0.99  -1.6  0.00470  1.6  98000 2.40  98000  98000  98000   2.50     1     1
x[13]  0.0029  0.0032 0.99  -1.6  0.00950  1.6  98000 2.40  98000  98000  98000   2.50     1     1
x[14] -0.0033  0.0032 0.99  -1.6 -0.00230  1.6  97000 2.40  97000  97000  97000   2.40     1     1
x[15]  0.0061  0.0032 0.99  -1.6  0.00520  1.6  95000 2.40  95000  95000  95000   2.40     1     1
x[16] -0.0006  0.0032 1.00  -1.7  0.00310  1.6 100000 2.50 100000  99000 100000   2.50     1     1
lp__  -8.0000  0.0230 2.80 -13.0 -7.70000 -3.9  15000 0.37  15000  14000  14000   0.36     1     1
      zRhat zsRhat zfsRhat zfsseff zfsreff tailseff tailreff medsseff medsreff madsseff madsreff
x[1]      1      1       1   16000    0.41    29000     0.72    82000     2.10    19000     0.48
x[2]      1      1       1   16000    0.41    30000     0.74    75000     1.90    19000     0.48
x[3]      1      1       1   16000    0.40    29000     0.72    79000     2.00    19000     0.47
x[4]      1      1       1   16000    0.40    29000     0.73    81000     2.00    19000     0.48
x[5]      1      1       1   17000    0.42    30000     0.75    80000     2.00    20000     0.50
x[6]      1      1       1   17000    0.41    28000     0.71    79000     2.00    20000     0.49
x[7]      1      1       1   17000    0.43    29000     0.73    82000     2.00    20000     0.49
x[8]      1      1       1   16000    0.41    30000     0.75    79000     2.00    19000     0.47
x[9]      1      1       1   16000    0.40    29000     0.72    81000     2.00    19000     0.47
x[10]     1      1       1   16000    0.41    29000     0.74    77000     1.90    19000     0.48
x[11]     1      1       1   16000    0.40    28000     0.71    79000     2.00    18000     0.46
x[12]     1      1       1   15000    0.38    29000     0.73    82000     2.00    19000     0.47
x[13]     1      1       1   15000    0.38    27000     0.69    81000     2.00    18000     0.46
x[14]     1      1       1   16000    0.41    28000     0.69    78000     1.90    19000     0.48
x[15]     1      1       1   17000    0.42    29000     0.73    80000     2.00    20000     0.49
x[16]     1      1       1   16000    0.41    30000     0.74    82000     2.10    19000     0.47
lp__      1      1       1   21000    0.54    20000     0.49    17000     0.43    24000     0.59

The Bulk-ESS for all \(x\) is larger than 9.522310^{4}. However tail-ESS for all \(x\) is less than 2.99310^{4}. Further, bulk-ESS for lp__ is only 1.448910^{4}.
If we take a look at all the Stan examples in this notebook, we see that the bulk-ESS for lp__ is always below 0.5. This is because lp__ correlates strongly with the total energy in HMC, which is sampled using a random walk proposal once per iteration. Thus, it’s likely that lp__ has some random walk behavior, as well, leading to autocorrelation and a small relative efficiency. At the same time, adaptive HMC can create antithetic Markov chains which have negative auto-correlations at odd lags. This results in a bulk-ESS greater than S for some parameters.

Let’s check the effective sample size in different parts of the posterior by computing the effective sample size for small interval estimates for x[1].

plot_local_ess(fit_n, par = 1, nalpha = 100)

The effective sample size for probability estimate for a small interval is close to 1 with a slight drop in the tails. This is a good result, but far from the effective sample size for the bulk, mean, and median estimates. Let’s check the effective sample size for quantiles.

plot_quantile_ess(fit = fit_n, par = 1, nalpha = 100)

Central quantile estimates have higher effective sample size than tail quantile estimates.

The total energy of HMC should affect how far in the tails a chain in one iteration can go. Fat tails of the target have high energy, and thus only chains with high total energy can reach there. This will suggest that the random walk in total energy would cause random walk in the variance of \(x\). Let’s check the second moment of \(x\).

samp_x2 <- as.array(fit_n, pars = "x")^2
monitor(samp_x2)

Inference for the input samples (4 chains: each with iter = 10000; warmup = 0):

          Q5  Q50 Q95 Mean  SD  Rhat Bulk_ESS Tail_ESS
x[1]  0.0042 0.46 3.8 1.00 1.4     1    16443    18225
x[2]  0.0037 0.44 3.8 0.99 1.4     1    16492    19392
x[3]  0.0040 0.45 3.8 0.98 1.4     1    16148    18342
x[4]  0.0039 0.45 3.9 1.00 1.5     1    16070    18288
x[5]  0.0042 0.45 3.9 1.00 1.4     1    16785    18672
x[6]  0.0041 0.45 3.9 1.00 1.4     1    16572    17525
x[7]  0.0040 0.45 3.7 0.99 1.4     1    17097    19120
x[8]  0.0039 0.46 3.8 1.00 1.4     1    16397    18152
x[9]  0.0039 0.45 3.8 1.00 1.4     1    15922    18049
x[10] 0.0041 0.44 3.7 0.98 1.4     1    16461    18098
x[11] 0.0043 0.46 3.9 1.00 1.4     1    16008    19463
x[12] 0.0038 0.45 3.8 0.99 1.4     1    15368    17674
x[13] 0.0037 0.44 3.8 0.98 1.4     1    15371    16755
x[14] 0.0038 0.45 3.8 0.98 1.4     1    16461    17715
x[15] 0.0040 0.45 3.8 0.98 1.4     1    16655    19241
x[16] 0.0043 0.47 3.9 1.00 1.4     1    16400    19741

For each parameter, Bulk_ESS and Tail_ESS are crude measures of 
effective sample size for bulk and tail quantities respectively (good values is 
ESS > 400), and Rhat is the potential scale reduction factor on rank normalized
split chains (at convergence, Rhat = 1).

res <- monitor_extra(samp_x2)
print(res)

Inference for the input samples (4 chains: each with iter = 10000; warmup = 0):

      mean se_mean  sd     Q5  Q50 Q95  seff reff sseff zseff zsseff zsreff  Rhat sRhat zRhat
x[1]  1.00   0.012 1.4 0.0042 0.46 3.8 15000 0.37 15000 16000  16000   0.41     1     1     1
x[2]  0.99   0.011 1.4 0.0037 0.44 3.8 16000 0.39 16000 16000  16000   0.41     1     1     1
x[3]  0.98   0.011 1.4 0.0040 0.45 3.8 15000 0.37 15000 16000  16000   0.40     1     1     1
x[4]  1.00   0.012 1.5 0.0039 0.45 3.9 15000 0.36 15000 16000  16000   0.40     1     1     1
x[5]  1.00   0.012 1.4 0.0042 0.45 3.9 15000 0.38 15000 17000  17000   0.42     1     1     1
x[6]  1.00   0.012 1.4 0.0041 0.45 3.9 14000 0.36 14000 17000  17000   0.41     1     1     1
x[7]  0.99   0.011 1.4 0.0040 0.45 3.7 15000 0.39 15000 17000  17000   0.43     1     1     1
x[8]  1.00   0.012 1.4 0.0039 0.46 3.8 15000 0.37 15000 16000  16000   0.41     1     1     1
x[9]  1.00   0.012 1.4 0.0039 0.45 3.8 13000 0.34 13000 16000  16000   0.40     1     1     1
x[10] 0.98   0.011 1.4 0.0041 0.44 3.7 15000 0.37 15000 16000  16000   0.41     1     1     1
x[11] 1.00   0.011 1.4 0.0043 0.46 3.9 15000 0.38 15000 16000  16000   0.40     1     1     1
x[12] 0.99   0.012 1.4 0.0038 0.45 3.8 14000 0.36 14000 15000  15000   0.38     1     1     1
x[13] 0.98   0.012 1.4 0.0037 0.44 3.8 14000 0.34 14000 15000  15000   0.38     1     1     1
x[14] 0.98   0.011 1.4 0.0038 0.45 3.8 15000 0.36 15000 16000  16000   0.41     1     1     1
x[15] 0.98   0.011 1.4 0.0040 0.45 3.8 15000 0.39 15000 17000  17000   0.42     1     1     1
x[16] 1.00   0.011 1.4 0.0043 0.47 3.9 16000 0.39 16000 16000  16000   0.41     1     1     1
      zsRhat zfsRhat zfsseff zfsreff tailseff tailreff medsseff medsreff madsseff madsreff
x[1]       1       1   18000    0.46    18000     0.46    19000     0.48    23000     0.58
x[2]       1       1   20000    0.49    19000     0.48    19000     0.48    25000     0.62
x[3]       1       1   19000    0.46    18000     0.46    19000     0.47    24000     0.60
x[4]       1       1   19000    0.47    18000     0.46    19000     0.48    24000     0.60
x[5]       1       1   20000    0.49    19000     0.47    20000     0.50    25000     0.63
x[6]       1       1   18000    0.44    18000     0.44    20000     0.49    23000     0.57
x[7]       1       1   19000    0.48    19000     0.48    20000     0.49    23000     0.58
x[8]       1       1   19000    0.47    18000     0.45    19000     0.47    24000     0.60
x[9]       1       1   18000    0.46    18000     0.45    19000     0.47    22000     0.55
x[10]      1       1   19000    0.46    18000     0.45    19000     0.48    24000     0.60
x[11]      1       1   19000    0.49    19000     0.49    18000     0.46    25000     0.62
x[12]      1       1   19000    0.47    18000     0.44    19000     0.46    24000     0.60
x[13]      1       1   19000    0.46    17000     0.42    18000     0.46    24000     0.60
x[14]      1       1   19000    0.47    18000     0.44    19000     0.48    23000     0.59
x[15]      1       1   20000    0.49    19000     0.48    20000     0.49    24000     0.61
x[16]      1       1   20000    0.50    20000     0.49    19000     0.47    24000     0.61

The mean of the bulk-ESS for \(x_j^2\) is 1.62906210^{4}, which is quite close to the bulk-ESS for lp__. This is not that surprising as the potential energy in normal model is proportional to \(\sum_{j=1}^J x_j^2\).

Let’s check the effective sample size in different parts of the posterior by computing the effective sample size for small interval probability estimates for x[1]^2.

plot_local_ess(fit = samp_x2, par = 1, nalpha = 100)

The effective sample size is mostly a bit below 1, but for the right tail of \(x_1^2\) the effective sample size drops. This is likely due to only some iterations having high enough total energy to obtain draws from the high energy part of the tail. Let’s check the effective sample size for quantiles.

plot_quantile_ess(fit = samp_x2, par = 1, nalpha = 100)

We can see the correlation between lp__ and magnitude of x[1] in the following plot.

samp <- as.array(fit_n)
qplot(
  as.vector(samp[, , "lp__"]),
  abs(as.vector(samp[, , "x[1]"]))
) + 
  labs(x = 'lp__', y = 'x[1]')

Low lp__ values corresponds to high energy and more variation in x[1], and high lp__ corresponds to low energy and small variation in x[1]. Finally \(\sum_{j=1}^J x_j^2\) is perfectly correlated with lp__.

qplot(
  as.vector(samp[, , "lp__"]),
  as.vector(apply(samp[, , 1:16]^2, 1:2, sum))
) + 
  labs(x = 'lp__', y = 'sum(x^2)')

This shows that even if we get high effective sample size estimates for central quantities (like mean or median), it is important to look at the relative efficiency of scale and tail quantities, as well. The effective sample size of lp__ can also indicate problems of sampling in the tails.

Original Computing Environment

makevars <- file.path(Sys.getenv("HOME"), ".R/Makevars")
if (file.exists(makevars)) {
  writeLines(readLines(makevars)) 
}

CXX14FLAGS=-O3 -Wno-unused-variable -Wno-unused-function
CXX14 = $(BINPREF)g++ -m$(WIN) -std=c++1y
CXX11FLAGS=-O3 -Wno-unused-variable -Wno-unused-function

devtools::session_info("rstan")

- Session info -----------------------------------------------------------------------------------
 setting  value                       
 version  R version 3.5.1 (2018-07-02)
 os       Windows 10 x64              
 system   x86_64, mingw32             
 ui       RTerm                       
 language (EN)                        
 collate  German_Germany.1252         
 ctype    German_Germany.1252         
 tz       Europe/Berlin               
 date     2018-12-20                  

- Packages ---------------------------------------------------------------------------------------
 package      * version    date       lib source                          
 assertthat     0.2.0      2017-04-11 [1] CRAN (R 3.5.0)                  
 backports      1.1.3      2018-12-14 [1] CRAN (R 3.5.1)                  
 BH             1.66.0-1   2018-02-13 [1] CRAN (R 3.5.0)                  
 callr          3.1.0      2018-12-10 [1] CRAN (R 3.5.1)                  
 cli            1.0.1      2018-09-25 [1] CRAN (R 3.5.1)                  
 colorspace     1.3-2      2016-12-14 [1] CRAN (R 3.5.0)                  
 crayon         1.3.4      2017-09-16 [1] CRAN (R 3.5.0)                  
 desc           1.2.0      2018-05-01 [1] CRAN (R 3.5.0)                  
 digest         0.6.18     2018-10-10 [1] CRAN (R 3.5.1)                  
 fansi          0.4.0      2018-10-05 [1] CRAN (R 3.5.1)                  
 ggplot2      * 3.1.0      2018-10-25 [1] CRAN (R 3.5.1)                  
 glue           1.3.0      2018-07-17 [1] CRAN (R 3.5.1)                  
 gridExtra    * 2.3        2017-09-09 [1] CRAN (R 3.5.0)                  
 gtable         0.2.0      2016-02-26 [1] CRAN (R 3.5.0)                  
 inline         0.3.15     2018-05-18 [1] CRAN (R 3.5.1)                  
 labeling       0.3        2014-08-23 [1] CRAN (R 3.5.0)                  
 lattice        0.20-35    2017-03-25 [2] CRAN (R 3.5.1)                  
 lazyeval       0.2.1      2017-10-29 [1] CRAN (R 3.5.0)                  
 loo            2.0.0      2018-04-11 [1] CRAN (R 3.5.1)                  
 magrittr       1.5        2014-11-22 [1] CRAN (R 3.5.0)                  
 MASS           7.3-50     2018-04-30 [2] CRAN (R 3.5.1)                  
 Matrix         1.2-14     2018-04-13 [2] CRAN (R 3.5.1)                  
 matrixStats    0.54.0     2018-07-23 [1] CRAN (R 3.5.1)                  
 mgcv           1.8-26     2018-11-21 [1] CRAN (R 3.5.1)                  
 munsell        0.5.0      2018-06-12 [1] CRAN (R 3.5.1)                  
 nlme           3.1-137    2018-04-07 [2] CRAN (R 3.5.1)                  
 pillar         1.3.1      2018-12-15 [1] CRAN (R 3.5.1)                  
 pkgbuild       1.0.2      2018-10-16 [1] CRAN (R 3.5.1)                  
 plyr           1.8.4      2016-06-08 [1] CRAN (R 3.5.0)                  
 prettyunits    1.0.2      2015-07-13 [1] CRAN (R 3.5.1)                  
 processx       3.2.1      2018-12-05 [1] CRAN (R 3.5.1)                  
 ps             1.2.1      2018-11-06 [1] CRAN (R 3.5.1)                  
 R6             2.3.0      2018-10-04 [1] CRAN (R 3.5.1)                  
 RColorBrewer   1.1-2      2014-12-07 [1] CRAN (R 3.5.0)                  
 Rcpp           1.0.0      2018-11-07 [1] CRAN (R 3.5.1)                  
 RcppEigen      0.3.3.5.0  2018-11-24 [1] CRAN (R 3.5.1)                  
 reshape2       1.4.3      2017-12-11 [1] CRAN (R 3.5.0)                  
 rlang          0.3.0.1    2018-10-25 [1] CRAN (R 3.5.1)                  
 rprojroot      1.3-2      2018-01-03 [1] CRAN (R 3.5.0)                  
 rstan        * 2.18.2     2018-11-07 [1] CRAN (R 3.5.1)                  
 scales         1.0.0      2018-08-09 [1] CRAN (R 3.5.1)                  
 StanHeaders  * 2.18.0-1   2018-12-13 [1] CRAN (R 3.5.1)                  
 stringi        1.2.4      2018-07-20 [1] CRAN (R 3.5.1)                  
 stringr      * 1.3.1      2018-05-10 [1] CRAN (R 3.5.1)                  
 tibble       * 1.4.2      2018-01-22 [1] CRAN (R 3.5.0)                  
 utf8           1.1.4      2018-05-24 [1] CRAN (R 3.5.1)                  
 viridisLite    0.3.0      2018-02-01 [1] CRAN (R 3.5.0)                  
 withr          2.1.2.9000 2018-12-18 [1] Github (jimhester/withr@be57595)

[1] C:/Users/paulb/Documents/R/win-library/3.5
[2] C:/Program Files/R/R-3.5.1/library

Rank-normalized split-Rhat and effective sample size

Aki Vehtari, Andrew Gelman, Daniel Simpson, Bob Carpenter, Paul Bürkner

First version 2017-05-23. Last modified 2018-12-20.

1 Introduction

2 Review of split-\(\widehat{R}\) and effective sample size

2.1 Split-\(\widehat{R}\)

2.2 Effective sample size

2.2.1 Estimation of the Effective Sample Size

3 Rank normalized split-\(\widehat{R}\) and effective sample size

3.1 Rank normalized split-\(\widehat{R}\)

3.2 Rank normalized folded-split-\(\widehat{R}\)

3.3 Effective sample size using rank normalized values

3.4 Efficiency estimates for the cumulative distribution function

3.5 Efficiency estimates for quantiles and tail-ESS

3.6 Efficiency estimates for median and MAD

3.7 Monte Carlo error estimates for quantiles

3.8 Efficiency estimate for small interval probability estimates

3.9 Rank plots

3.10 Proposed changes in Stan

3.11 Proposed additions to bayesplot

3.12 Warning thresholds

4 Examples

4.1 Cauchy: A distribution with infinite mean and variance

4.1.1 Nominal parameterization of Cauchy

4.1.1.1 Default Stan options

4.1.2 Alternative parameterization of Cauchy

4.1.3 Half-Cauchy with nominal parameterization

4.2 Hierarchical model: Eight Schools

4.2.1 A Centered Eight Schools model

4.2.1.1 Centered Eight Schools model

4.2.2 Non-centered Eight Schools model

References

Appendices

Appendix A: Abbreviations

Appendix B: Examples of rank normalization

Appendix C: Variance of the cumulative distribution function

Appendix D: Normal distributions with additional trend, shift or scaling

Adding the same trend to all chains

Shifting one chain

Scaling one chain

Appendix E: Cauchy: A distribution with infinite mean and variance

Nominal parameterization of Cauchy

Default Stan options

Default Stan options + increased maximum treedepth

Default Stan options + increased maximum treedepth + longer chains

First alternative parameterization of the Cauchy distribution

Another alternative parameterization of the Cauchy distribution

Half-Cauchy distribution with nominal parameterization

Alternative parameterization of the half-Cauchy distribution

The Cauchy distribution with Jags

Appendix F: Hierarchical model: Eight Schools

A Centered Eight Schools model

Centered parameterization with longer chains

Centered parameterization with very long chains

Centered parameterization with very long chains and thinning

Non-centered Eight Schools model

Non-centered parameterization with default MCMC options

Eight Schools with Jags

Centered Eight Schools Model

Non-Centered Eight Schools Model

Appendix G: Dynamic HMC and effective sample size

Original Computing Environment